HAMILTONIAN 2-FORMS IN KÄHLER GEOMETRY, III EXTREMAL METRICS AND STABILITY

VESTISLAV APOSTOLOV, DAVID M. J. CALDERBANK, PAUL GAUDUCHON, AND CHRISTINA W. TØNNESEN-FRIEDMAN

Abstract. This paper concerns the existence and explicit construction of extremal Kähler metrics on total spaces of projective bundles, which have been studied in many places. We present a unified approach, motivated by the theory of hamiltonian 2-forms (as introduced and studied in previous papers in the series) but this paper is largely independent of that theory.

We obtain a characterization, on a large family of projective bundles, of the ‘admissible’ Kähler classes (i.e., those compatible with the bundle structure in a way we make precise) which contain an extremal Kähler metric. In many cases every Kähler class is admissible. In particular, our results complete the classification of extremal Kähler metrics on geometrically ruled surfaces, answering several long-standing questions.

We also find that our characterization agrees with a notion of K-stability for admissible Kähler classes. Our examples and nonexistence results therefore provide a fertile testing ground for the rapidly developing theory of stability for projective varieties, and we discuss some of the ramifications. In particular we obtain examples of projective varieties which are destabilized by a non-algebraic degeneration.

In this paper we give a systematic overview of some explicit constructions of extremal Kähler metrics on projective bundles and relate our constructions to the theory of stability for algebraic varieties.

There has been a great deal of interest recently in the relation between extremal Kähler metrics, especially constant scalar curvature (CSC) Kähler metrics, and stability: initiated by ideas going back to Yau [56], there are conjectures [53, 15] that the existence of an extremal or CSC Kähler metric, in an integral Kähler class Ω on a compact complex manifold M, should be equivalent to an algebraic geometric stability condition on the Kodaira embeddings of M into P (H^{0}(M, L^{k})^{∗}) for k ≫ 1, where c_{1}(L) = Ω/2π.

Our own interest in constructions of extremal Kähler metrics has been renewed and stimulated not only by these developments, but also by a unifying principle, which we shall explain and apply here, underlying explicit examples of such metrics on projective bundles obtained in [8, 27, 30, 31, 34, 47, 54]. In short these examples have in common the presence of a hamiltonian 2-form of order 1.

The general theory of hamiltonian 2-forms has been developed in our previous papers [4, 5]. A main feature of the Kähler metrics admitting a hamiltonian 2-form is the appearance of hidden symmetries, which, in the compact case, results in a rigid hamiltonian torus action of dimension the order of the hamiltonian 2-form.

Date: March 2007. The first author was supported by an NSERC Discovery Grant, the second author by an EPSRC

Advanced Research Fellowship and the fourth author by the Union College Faculty Research Fund.

2 V. APOSTOLOV, D. CALDERBANK, P. GAUDUCHON, AND C. TØNNESEN-FRIEDMAN

Motivated by a non-existence result proved in Appendix A.2, in this paper we focus our attention to the case when this order is 1. Under some mild additional hypotheses, the resulting manifolds are biholomorphic to projective bundles of the form P (E_{0} ⊕ E_{∞}) → S, where E_{0}, E_{∞} are projectively flat hermitian holomorphic vector bundles over a compact Kähler manifold S. The latter is locally a Kähler product ∏_{a∈A} S_{a} where A is the finite set of distinct constant eigenvalues of the hamitonian 2-form, η_{a} say, and each S_{a} has complex dimension equal to the multiplicity of η_{a}; furthermore, for each a in A there is an associated 2-form ω_{a} on S (with ±ω_{a} being the component of the Kähler form of S on S_{a} and ± is chosen so that η_{a} ω_{a} is positive definite) such that c_{1}(E_{∞})/rk(E_{∞}) − c_{1}(E_{0})/rk(E_{0}) = ∑_{a∈A}[ω_{a}/2π].

Complex projective bundles of this form will be referred to as admissible bundles or admissible manifolds. Likewise, the Kähler metrics, forms and classes arising on admissible manifolds according to the general theory of hamiltonian 2-forms will be called admissible. By their very definition, admissible Kähler metrics admit a natural isometric hamiltonian S^{1}-action whose generator is denoted by K. Section 1 is mainly devoted to make precise the above notions and conventions.

As in the first constructions of extremal Kähler metrics by E. Calabi in [7], the search for admissible extremal Kähler metrics in a given admissible class Ω on an admissible 2m-manifold involves a polynomial of degree equal to or less than m+2, here denoted by F_{Ω}, which we call the extremal polynomial of Ω. This polynomial plays a pivotal role in the whole paper. Our first main result can be stated as follows.

Theorem 1. Let M = P (E_{0} ⊕E_{∞}) → S be an admissible manifold, where the base S is a local Kähler product of CSC metrics (±g_{a}, ±ω_{a}). Then there is an extremal Kähler metric in an admissible Kähler class Ω if and only if the extremal polynomial F_{Ω} is positive on (−1, 1). This condition always holds if Ω is ‘sufficiently small’; if it does, there is an admissible extremal Kähler metric in Ω, which is CSC if and only if the Futaki invariant 𝔉_{Ω}(K) vanishes (i.e., F_{Ω} has degree ≤ m + 1).

The admissible Kähler classes containing an extremal Kähler metric form a nonempty open subset of all such classes, and those containing a CSC Kähler metric form a real analytic hypersurface which is nonempty if c_{1}(E_{∞})/rk(E_{∞}) − c_{1}(E_{0})/rk(E_{0}) is strictly indefinite (i.e., the definite forms ω_{a} on S_{a} do not all have the same sign).

Geometrically, an admissible Kähler class is sufficiently small if the base S is large (low curvature) compared to the fibres (high curvature). Thus the above theorem asserts the existence of extremal Kähler metrics with curvature concentrated in the fibres, which agrees with the results in [32, 45]. Note also that by the uniqueness theorem of Chen–Tian [10, 11], ‘the’ extremal Kähler metric in Ω (if it exists) is unique up to automorphism.

Section 2 is devoted to the proof of Theorem 1. This involves several new ingredients of independent interest and relies in a crucial way on a recent results of Chen– Tian [10, 11], quoted in this paper as Theorem 4. First of all, we consider what implications the existence of an extremal Kähler metric has for admissible bundles, Kähler classes and metrics. In §2.1, we apply the Matsushima–Lichnerowicz criterion [36, 41] to obtain information about the automorphism group of an admissible bundle when the base metric on S is CSC. In §2.2, we use this information to show that, for any admissible Kähler class Ω, the so-called extremal vector field of Ω is a multiple of the canonical hamiltonian Killing vector field K and we compute the Futaki invariant 𝔉_{Ω}(K). In §2.3, we provide an explicit formula for the Mabuchi– Guan–Simanca (modified) K-energy functional relative to the natural S^{1}-action on the space of admissible Kähler metrics in Ω, and show that it is determined by the extremal polynomial F_{Ω} which has as a leading coefficient a nonzero multiple of 𝔉_{Ω}(K). By using the Chen–Tian Theorem 4, we deduce that if Ω contains an extremal Kähler metric, then F_{Ω} must be nonnegative on (−1, 1). In the boundary case, when F_{Ω} is nonnegative with zeroes on (−1, 1), we show non-existence of extremal Kähler metrics in Ω via a delicate limiting argument involving the LeBrun– Simanca stability theorem [35], the general theory of hamiltonian 2-forms [5], and the uniqueness of extremal Kähler metrics modulo automorphisms established in [11].

In §2.4, we apply Proposition 1 of section 1 to give a constructive proof of the existence part of Theorem 1, unifying and generalizing work of Calabi, Guan, Hwang, Hwang–Singer and the fourth author [8, 27, 30, 31, 54].

In section 3 we present further existence and nonexistence results for extremal and CSC metrics by computing the extremal polynomial on various examples and testing its positivity on (−1, 1). In many of these examples, every Kähler class on M is admissible (see Remark 2) and, therefore, Theorem 1 describes precisely which Kähler classes do contain an extremal Kähler metric. This is the case when M = P (O ⊕ L) → S is an admissible geometrically ruled complex surface (meaning here that the degree of L is non-zero); the extremal polynomial F_{Ω}(z) is then a quartic divisible by 1 − z^{2}. We thus obtain a complete resolution of the existence question for extremal Kähler metrics on these complex surfaces. This, combined with an observation from [2], fills in the missing step in the complete classification of extremal Kähler metrics on all geometrically complex ruled surfaces, which can be specifically stated as follows.

Corollary 1. Let M = P (E) → S be a geometrically ruled complex surface, where E is a rank 2 holomorphic vector bundle over a compact Riemann surface S of genus g. If M admits an extremal Kähler metric, then one of the following two cases holds:

• E is polystable: then M = P (E) admits a locally-symmetric CSC Kähler in each Kähler class;

• M = P (O ⊕ L) → S is admissible, the Kähler cone is a cone on an open interval (a, b); the extremal Kähler metrics are precisely those of [7, 30, 54], which are admissible and locally cohomogeneity one, with Kähler classes parameterized by a cone on a subinterval (a, c), with c = b if and only if S has genus 0 or 1.

We refer the reader to [2] for more details. We also note that Ross–Thomas [45] and G. Székelyhidi [50, 51] independently obtained similar results for rational Kähler classes on ruled surfaces, by using a different approach relying on recent works of S. Donaldson [15, 17, 18].

Our results provide a fertile testing ground for the conjectures relating extremal and CSC Kähler metrics to stability, and we explore this in section 4. In §4.2 we relate our results to those of Ross–Thomas [45] and Hong [32]: in particular, we show that there are CSC metrics on projective bundles P (E) → S for which E is only (slope) polystable with respect one Kähler class on S up to scale.

In §4.4 we relate Theorem 1 to the notions of K-polystability [15, 52] or relative K-polystability [50] for Kähler classes, which are conjectured to be equivalent to the existence of a CSC or extremal Kähler metric in a given class. Actually, to be precise, we use a closely related notion of (relative) slope K-polystability defined by Ross–Thomas in [45], which is stronger than just specializing the notion of Kpolystability in the sense [15] to certain test configurations: the notion of (relative) slope K-polystability used here might more properly be referred to as analytic (relative) slope K-polystability, and is the notion relevant to assigning a sign of the Futaki invariant on analytic test configurations (while the original definitions of [53, 15] require only algebraic test configurations). By generalizing a beautiful calculation of G. Székelyhidi [50] for ruled surfaces, we establish the following result.

Theorem 2. Let Ω be an admissible integral Kähler class on M = P (E_{0} ⊕ E_{∞}) → S, where S is CSC. If Ω is slope K-polystable, it contains a CSC Kähler metric, and if it is relatively slope K-polystable, it contains an extremal Kähler metric.

It is natural to ask if (relative) K-polystability in the sense of [15, 50] implies the existence of a CSC (or extremal) Kähler metric in Ω. We find that this is true if dim S ≤ 4, but for dim S ≥ 6, we are only able to show that the extremal polynomial is positive on (−1, 1) ∩ ℚ.

Before our work, it was believed that K-polystability implies slope K-polystability in general, but the proof in [45, 46] only shows that it implies slope K-semistability, the gap being closely related to the issue of positivity (versus nonnegativity) of the extremal polynomial at irrational points in (−1, 1).

To show that this is a genuine problem, we end with some examples, on projective line bundles over a product of three Riemann surfaces, of integral admissible Kähler classes Ω such that F_{Ω} is positive on (−1, 1)∩ℚ but has an irrational repeated root in (−1, 1). We find these examples intriguing, since by Theorem 1, these Kähler classes do not contain an extremal Kähler metric, so they should be unstable. However, even though they are projective varieties, the degeneration that demonstrates this instability is not algebraic. While we cannot prove that there is no other algebraic test configuration which would detect this instability, it is difficult to imagine how such a test configuration could be constructed. Thus, presumably, our examples are algebraically (relative) K-polystable but analytically only (relative) K-semistable. This suggests that the non-algebraic degenerations implicit in the use of slope Kpolystability may be essential to relate stability to existence of CSC and extremal Kähler metrics. ^{1}

We would like to thank Claude LeBrun for helpful comments, Richard Thomas for discussions concerning [45, 46], and the referee for useful suggestions.

Contents

1. Admissible bundles and Kähler metrics 5 1.1. Hamiltonian 2-forms 5 1.2. Admissible projective bundles 7 1.3. Admissible Kähler classes and canonical metrics 8 1.4. Admissible metrics 10 1.5. The isometry Lie algebra 11 2. Admissible extremal Kähler metrics 12 2.1. Automorphisms and the Matsushima–Lichnerowicz obstruction 12 2.2. The Futaki invariant and extremal vector field 14 2.3. K-energy and the extremal polynomial 15 2.4. A characterization of extremal admissible Kähler classes 17

1Motivated by our examples, G. Székelyhidi introduced in [51] a stronger notion of uniform (relative) K-polystability for integral Kähler classes, which, in our setting, forces F_{Ω} to be positive on (−1, 1) and, therefore, implies the existence of an admissible extremal metric.

3. Existence and nonexistence results for extremal Kähler metrics 21 3.1. Constructing admissible extremal Kähler metrics 21 3.2. Extremal Kähler metrics over a Riemann surface 22 3.3. Nonexistence of CSC Kähler metrics over a Hodge 4-manifold 23 3.4. CSC Kähler metrics over a product of two Riemann surfaces 24

4. K-stability and admissible extremal Kähler metrics 26 4.1. Introduction to stability 26 4.2. Stable bundles and CSC Kähler metrics 29 4.3. Extremal Kähler metrics and relative K-polystability 31 4.4. Relative K-polystability of admissible projective bundles 32 Appendix A. Relation to previous papers 36 A.1. Summary of the classification 36 A.2. A nonexistence result for order 2 extremal Kähler metrics 38 Appendix B. Proof of Lemma 6 39 References 40

1. Admissible bundles and Kähler metrics

This section is devoted to introducing the main ingredients of the general theory of hamiltonian 2-forms which we will use in this paper, and giving the precise definitions of the geometric entities mentioned in the introduction.

1.1. Hamiltonian 2-forms. A hamiltonian 2-form [4] on a Kähler manifold (M, J, g, ω), of real dimension 2m > 2 is a real (1, 1)-form (i.e., a J-invariant 2-form) ϕ such that

2∇_{X}ϕ = d tr ϕ ∧ (JX)^{♭} − (Jd tr ϕ) ∧ X^{♭}

for all X ∈ T M (where X^{♭}(Y) = g(X, Y) for Y ∈ T M and tr ϕ = ⟨ω, ϕ⟩_{g}). The momentum polynomial of ϕ is then defined to be

p(t) := (−1)^{m} pf(ϕ − tω) = t^{m} − (tr ϕ) t^{m−1} + ⋯ + (−1)^{m} pf ϕ,

where the pfaffian is defined by ϕ ∧ ⋯ ∧ ϕ = (pf ϕ)ω ∧ ⋯ ∧ ω. The reason ϕ is called hamiltonian is that for any t ∈ ℝ, p(t) is a hamiltonian for a Killing vector field K(t) := J grad_{g} p(t) (this is not difficult to show [4, §2]). The integer ℓ = max_{x∈M} dim span{K(t)_{x} : t ∈ ℝ} (with 0 ≤ ℓ ≤ m) is called the order of ϕ.

We do not wish to impose the study of hamiltonian 2-forms on the reader of this paper, since we only need the general theory as motivation for the classes of complex manifolds and metrics that we shall study. We therefore now state a classification result which reduces the theory of hamiltonian 2-forms of order 1 to an Ansatz for metrics on projective bundles. This result follows easily from [4, 5], as we explain in Appendix A, where we also explain why we restrict attention to the order 1 case.

Theorem 3. Let (M, g, J, ω) be a compact connected Kähler 2m-manifold with a hamiltonian 2-form ϕ of order 1. Then:

• there is an effective isometric hamiltonian S^{1} action on M generated by a vector field K = J grad_{g} z such that the stable quotient of M by the induced holomorphic ℂ^{×} action is a compact connected complex manifold Ŝ of real dimension 2(m − 1); • without loss, the image of the momentum map z is [−1, 1], and there are Kähler manifolds S_{a} of dimension 2d_{a} and real numbers x_{a}, indexed by a ∈ Â ⊂ ℕ ∪ {∞}, such that Ŝ is covered by ∏_{a∈Â} S_{a} and 0 < |x_{a}| ≤ 1 with equality iff a ∈ {0, ∞};

• z is a Morse–Bott function [1] on M with critical set z^{−1}({−1, 1}), M^{0} := z^{−1}((−1, 1)) is a principal ℂ^{×}-bundle over Ŝ, and on each S_{a} there is a (1, 1)- form ω_{a} (which is the pullback of a tensor on Ŝ) such that either +ω_{a} or −ω_{a} is positive definite (with the sign chosen so that ω_{a}/x_{a} is positive definite), and thus gives rise to a Kähler metric on S_{a}, accordingly denoted by ±g_{a}, such that the Kähler structure on M^{0} is

(1) g = ∑_{a∈Â} \frac{1 + x_{a}z}{x_{a}} g_{a} + \frac{dz^{2}}{Θ(z)} + Θ(z)θ^{2}, ω = ∑_{a∈Â} \frac{1 + x_{a}z}{x_{a}} ω_{a} + dz ∧ θ,

where θ is a connection 1-form (θ(K) = 1) with dθ = ∑_{a∈Â} ω_{a}, and Θ is a smooth function on [−1, 1] satisfying

Θ > 0 on (−1, 1), (2)

Θ(±1) = 0, Θ′(±1) = ∓2; (3)

• if 0 ∈ Â then x_{0} = 1, S_{0} = ℂP^{d_{0}} and g_{0} is the Fubini–Study metric of scalar curvature 2d_{0}(d_{0} + 1), otherwise we set d_{0} = 0; likewise, if ∞ ∈ Â then x_{∞} = −1, S_{∞} = ℂP^{d_{∞}} and −g_{∞} is the Fubini–Study metric of scalar curvature 2d_{∞}(d_{∞} + 1), otherwise we set d_{∞} = 0; we also put A = Â ∖ {0, ∞} ⊂ ℤ^{+};

• the blow-up M̂ of M along z^{−1}({−1, 1}) is ℂ^{×}-equivariantly biholomorphic to M^{0} ×_{ℂ^{×}} ℂP^{1} → Ŝ and Ŝ is a fibre product of flat projective unitary ℂP^{d_{0}}- and ℂP^{d_{∞}}-bundles over a Kähler manifold S covered by ∏_{a∈A} S_{a}. \underline{If we assume that} Ŝ = P (E_{0}) ×_{S} P (E_{∞}) → S for projectively-flat hermitian vector bundles E_{0}, E_{∞} → S, then these bundles can be chosen so that M is ℂ^{×}- equivariantly biholomorphic to P (E_{0} ⊕ E_{∞}) → S, and we therefore have \overline{c}_{1}(E_{∞}) − \overline{c}_{1}(E_{0}) = ∑_{a∈A}[ω_{a}/2π], where \overline{c}_{1}(E) = c_{1}(E)/ rank E.

The final assumption of this theorem is automatic if π_{1}(S) = 1, when Ŝ = ℂP^{d_{0}} × S × ℂP^{d_{∞}} and there is a line bundle L → S with c_{1}(L) = ∑_{a∈A}[ω_{a}/2π] such that M is ℂ^{×}-equivariantly biholomorphic to P (O ⊗ℂ^{d_{0}+1} ⊕L⊗ℂ^{d_{∞}+1}) → S.

We shall use this theorem as an Ansatz for constructing extremal Kähler metrics on projective bundles of the form P (E_{0} ⊕ E_{∞}) → S. For this we shall use the following elementary computation [27, 30, 4].

Proposition 1. Let g be a Kähler metric of the form (1) on M^{0} and write F (z) = Θ(z)p_{c}(z) with p_{c}(z) := ∏_{a∈Â}(1+x_{a}z)^{d_{a}}. Then g is extremal, with Scal_{g} a constant affine function of z, iff

• there is a polynomial P of degree ≤ #Â + 1 such that

(4)  F′′(z) = ( ∏_{a∈Â}(1 + x_{a}z)^{d_{a}−1})P (z);

• for all a, ±g_{a} has constant scalar curvature Scal_{±g_{a}} = ±2d_{a}s_{a} where

(5)  P (−1/x_{a}) = 2d_{a}s_{a}x_{a} ∏_{b∈Â\\{a\}}(1 − \frac{x_{b}}{x_{a}} ).

The metric g then has constant scalar curvature iff P has degree ≤ #Â.

Compared to [4, 5], we have rescaled F (z) and p_{c}(z) by ∏_{a∈Â} x_{a}^{d_{a}}: this is convenient as p_{c}(z) is then positive on (−1, 1). Thus Θ is positive on (−1, 1) if and only if F is. Also if Θ(z) satisfies (3), then F (z) = Θ(z)p_{c}(z) satisfies

(6)  F (±1) = 0,  F′(±1) = ∓2p_{c}(±1).

In the following subsections we study metrics of the form (1) and show that the conditions of Theorem 3 are sufficient for the compactification of such metrics on a projective bundle M = P (E_{0} ⊕ E_{∞}) → S.

1.2. Admissible projective bundles. We use Theorem 3 (including the final assumption) as motivation for the class of compact complex manifolds we will study. A projective bundle of the form M = P (E_{0} ⊕ E_{∞}) →^{p} S will be called admissible or an admissible manifold if:

• S is a covered by a product S̃ = ∏_{a∈A} S_{a} (for A ⊂ ℤ^{+}) of simply-connected Kähler manifolds (S_{a}, ±g_{a}, ±ω_{a}) of real dimensions 2d_{a}, with (g_{a}, ω_{a}) being pullbacks of tensors on S; here and henceforth, with slight abuse of notation, the signs ± before the Kähler structure (±g_{a}, ±ω_{a}) on the complex manifold S_{a} mean that either +ω_{a} or −ω_{a} is a positive definite (1, 1)-form which defines a Kähler metric denoted by +g_{a} or −g_{a}, respectively. We will use the notation ±ω_{a} (rep. ±g_{a}) to denote the positive definite form (resp. the Riemannian metric) on S_{a}.

• E_{0} and E_{∞} are holomorphic projectively-flat hermitian vector bundles over S of ranks d_{0} + 1 and d_{∞} + 1 with \overline{c}_{1}(E_{∞}) − \overline{c}_{1}(E_{0}) = [ω_{S}/2π] and ω_{S} = ∑_{a∈A} ω_{a}.

The second condition (cf. [33]) means that we can fix hermitian metrics on E_{0} and E_{∞} whose Chern connections have tracelike curvatures Ω_{0} ⊗ Id_{E_{0}} and Ω_{∞} ⊗ Id_{E_{∞}} satisfying Ω_{∞} − Ω_{0} = ∑_{a∈A} ω_{a}. We normalize the induced fibrewise Fubini–Study metrics (g_{0}, ω_{0}) and (−g_{∞}, −ω_{∞}) on P (E_{0}) and P (E_{∞}) to have scalar curvatures 2d_{0}(d_{0} + 1) and 2d_{∞}(d_{∞} + 1).

We collect a few remarks and notations that we will use. We omit pullbacks by obvious projections in these remarks.

(i) We sometimes let the index a take values in ℕ ∪ {∞} by setting d_{a} = 0 for a ∉ A ∪ {0, ∞} (so that S_{a} is a point and ω_{a} = 0). This range will be assumed unless otherwise stated. We set Â:= {a : d_{a} > 0} so that A = Â ∩ ℤ^{+}.

(ii) The pullbacks of E_{0} and E_{∞} to S̃ are of the form E_{0} ⊗ ℂ^{d_{0}+1} and E_{∞} ⊗ ℂ^{d_{∞}+1}, where L := E_{0}^{−1} ⊗ E_{∞} = ⨂_{a∈A} L_{a} for line bundles L_{a} → S_{a} with c_{1}(L_{a}) = [ω_{a}/2π].

(iii) e_{0} := P (E_{0} ⊕ 0) and e_{∞} := P (0 ⊕ E_{∞}) denote the ‘zero’ and ‘infinity’ subbundles of M, covered by S_{0} × S̃ and S̃ × S_{∞}, where S_{0} = ℂP^{d_{0}} and S_{∞} = ℂP^{d_{∞}}.

(iv) The blow-up of M along e_{0} ∪e_{∞} is M̂ := P (O ⊕L̂) →^{p̂} Ŝ, where Ŝ = P (E_{0})×_{S} P (E_{∞}) → S and L̂ = O(1)_{E_{0}} ⊗ O(−1)_{E_{∞}}, using the (fibrewise) hyperplane and tautological line bundles; we have c_{1}(L̂) = [ωŜ/2π], where ωŜ = ∑_{a} ω_{a}. If d_{0} > 0 or d_{∞} > 0 we say a blow-down occurs.

(v) ê_{0} and ê_{∞} denote the zero and infinity sections of M̂. The pullback of L̂ to S_{0} × S̃ × S_{∞} is L_{0} ⊗ L ⊗ L_{∞}, where L_{0} = O(1) → S_{0} and L_{∞} = O(−1) → S_{∞}.

(vi) Ŝ has a family of local Kähler product metrics gŜ(z) with Kähler forms zωŜ + ∑_{a} ω_{a}/x_{a} and we set gŜ = gŜ(0). (Note that gŜ is not compatible with ωŜ—the latter is symplectic, but not a Kähler form in general.) We let g_{S}(z) and g_{S} = g_{S}(0) denote the induced local Kähler product metrics on S.

We summarize the set-up with the following diagram of bundles and a blow-up:

[figure: M̂ = P (O ⊕ L̂); Ŝ = P (E_{0}) ×_{S} P (E_{∞}); M = P (E_{0} ⊕ E_{∞}); S,]

the universal cover (omitting pullbacks) of this diagram being:

[figure: P (O ⊕ L̂); ℂP^{d_{0}} × S̃ × ℂP^{d_{∞}}; P (O ⊗ ℂ^{d_{0}+1} ⊕ L ⊗ ℂ^{d_{∞}+1}); S̃ = ∏_{a∈A} S_{a}.]

Remark 1. The existence of the line bundle L̂ → Ŝ with c_{1}(L̂) = [ωŜ/2π] implies that ωŜ is integral in the sense that [ωŜ/2π] is in the image of H^{2}(Ŝ, ℤ) in H^{2}(Ŝ, ℝ). When Ŝ is a global Kähler product (i.e., we can write M = P (O ⊗ ℂ^{d_{0}+1} ⊕ L ⊗ ℂ^{d_{∞}+1}) → S = ∏_{a∈A} S_{a}), this integrality condition means that each ω_{a} is integral, i.e., the compact manifolds (S_{a}, ±g_{a}, ±ω_{a}) are Hodge.

We write ω_{a} = q_{a}α_{a} for an integer q_{a} ≠ 0, where α_{a} is a primitive integral Kähler form on S_{a}, so that q_{a} is a nonzero integer with the same sign as (g_{a}, ω_{a}), and q_{0} = 1 and q_{∞} = −1. We now compare [ω_{a}/2π] to the first Chern class c_{1}(K_{a}^{−1}) = [ρ_{a}/2π] of the anticanonical bundle of S_{a}, by writing [ρ_{a}] = p_{a}[α_{a}] + [ρ_{a}]_{0}, for a rational number p_{a}, where [ρ_{a}]_{0} · [α_{a}]^{d_{a}−1} = 0. Since any line bundle P with first Chern class [α_{a}/2π] is ample, P^{d_{a}+1} ⊗ K_{a} is nef by a result of Fujita [25] (see also [12, Theorem 8.3]), from which it follows easily that p_{a} ≤ d_{a} + 1. If S_{a} is a Riemann surface of genus g_{a}, then p_{a} = 2(1 − g_{a}) ≤ 2.

We set s_{a} = p_{a}/q_{a}. When ±g_{a} is CSC, we have Scal_{±g_{a}} = ±2d_{a}s_{a}, where the sign is that of q_{a}, so the scalar curvature of ±g_{a} has the same sign as p_{a}. Thus, in the case of a CSC Hodge manifold S_{a}, the Fujita inequality p_{a} ≤ d_{a} + 1 is (since |q_{a}| ≥ 1) equivalent to Scal_{±g_{a}} ≤ 2d_{a}(d_{a} + 1).

The conditions of Theorem 3 are also sufficient for the compactification of metrics of the form (1) on an admissible projective bundle M = P (E_{0} ⊕ E_{∞}) → S, where z: M → [−1, 1] with e_{0} = z^{−1}(1) and e_{∞} = z^{−1}(−1), and θ is a connection 1-form. Before discussing this, we introduce the Kähler classes to which they belong.

1.3. Admissible Kähler classes and canonical metrics. Suppose that M = P (E_{0} ⊕ E_{∞}) → S is an admissible bundle. We say that a Kähler class Ω on M is admissible if there are constants x_{a}, with x_{0} = 1, x_{∞} = −1, such that the pullback of Ω to M̂ has the form

∑_{a}[ω_{a}]/x_{a} + Ξ̂

up to scale, where the 2-forms ω_{a} are viewed as pullbacks to M̂ of the corresponding forms on Ŝ (induced by the local product Kähler structure ∏_{a} S_{a}) and Ξ̂ is Poincaré dual to 2π[ê_{0} + ê_{∞}]. Thus Ξ̂ = 2πc_{1}(V M̂), where V M̂ = O(2)_{O⊕L̂} ⊗ p̂^{∗}L̂ and O(−1)_{O⊕L̂} is the (fibrewise) tautological bundle of M̂ = P (O⊕L̂). (The first Chern class [ωŜ/2π] of L̂ itself pulls back to M̂ to give the Poincaré dual of [ê_{0} − ê_{∞}].)

It follows that admissible Kähler classes have the form

Ω = ∑_{a∈A}[ω_{a}]/x_{a} + Ξ

up to scale, where the pullback of Ξ to M̂ is [ω_{0}] − [ω_{∞}] + Ξ̂. Since pullback to a blow-up is injective on cohomology, admissible Kähler classes on M are uniquely determined by the parameters x_{a}.

If (g, ω) is any Kähler metric on M of the form (1) on M^{0}, then we claim Ω = [ω] is admissible. For this we first note that on M^{0}, the Kähler form ω is a linear combination ∑_{a∈A} ω_{a}/x_{a} + η, where

η = (z + 1)ω_{0} + ∑_{a∈A} zω_{a} + (z − 1)ω_{∞} + dz ∧ θ.

Here ω_{0}, ω_{∞} and θ are defined only on M^{0}. However, for a ∈ A, ω_{a} extends to a closed 2-form on M (as a pullback from S), so η is globally defined and closed on M (since ω is). The pullback of η to M̂ may be written ω_{0} − ω_{∞} + η̂ with η̂ = d(zθ) on M^{0}, and since ω_{0} and ω_{∞} are well-defined and closed on M̂ (as pullbacks from Ŝ), so is η̂, and we easily see^{2} that [η̂] = Ξ̂.

Observe that η depends implicitly on the choice of metric (g, ω) on M because the momentum map z does. However, the above shows that the cohomology class [η] is Ξ, independent of this choice. From this realisation of Ξ, it follows easily, by pulling back to e_{0} and e_{∞}, that for a cohomology class of the form ∑_{a∈A}[ω_{a}]/x_{a}+Ξ to be a Kähler class, it is necessary that for a ∈ A, 0 < |x_{a}| < 1 with the sign of x_{a} such that ω_{a}/x_{a} is positive. Conversely, we claim that any cohomology class of this form (with 0 < |x_{a}| < 1 and ω_{a}/x_{a} positive for a ∈ A) is an admissible Kähler class and contains a Kähler metric of the form (1) on M^{0} up to scale. To do this we construct a distinguished Kähler metric in each such class.

Let r_{0} and r_{∞} be the norm functions induced by the hermitian metrics on E_{0} and E_{∞}. Then z_{0} = \frac{1}{2}r_{0}^{2} and z_{∞} = \frac{1}{2}r_{∞}^{2} are fibrewise momentum maps for the U(1) actions given by scalar multiplication in E_{0} and E_{∞}, generated by K_{0} and K_{∞}. We equip M with a fibrewise Fubini–Study metric (g_{M/S}, ω_{M/S}): with our normalization of g_{0} and g_{∞}, each fibre is the Kähler quotient of the corresponding fibre of E_{0} ⊕ E_{∞} by the diagonal U(1) action at momentum level z_{0} + z_{∞} = 2; then on this momentum level, the function z = z_{0} − 1 = 1 − z_{∞} descends to a fibrewise momentum map M → [−1, 1] for the quotient U(1) action.

We extend (g_{M/S}, ω_{M/S}) to T M by requiring that the horizontal distribution of the induced connection on M is degenerate. To obtain a nondegenerate metric, we then set

g_{c} = ∑_{a∈A} \frac{1 + x_{a}z}{x_{a}} g_{a} + g_{M/S}, ω_{c} = ∑_{a∈A} \frac{1 + x_{a}z}{x_{a}} ω_{a} + ω_{M/S},

where the (g_{a}, ω_{a}) are pulled back from S; g_{c} is then a positive definite Kähler metric with respect to the canonical complex structure of M = P (E_{0} ⊕ E_{∞}) by the assumptions on the parameters x_{a}. We refer to (g_{c}, ω_{c}) as the canonical Kähler metric on M in the given admissible Kähler class. Lemma 1. For any 0 < |x_{a}| < 1 (a ∈ A), the corresponding canonical Kähler metric on M is of the form (1) on M^{0}, where Θ = Θ_{c} and Θ_{c}(z) = 1 − z^{2}. Proof. The inverse image in E_{0} ⊕ E_{∞} of M^{0} = M \ (z^{−1}(−1) ∪ z^{−1}(1)) may be viewed as an open subset of O(−1)_{E_{0}} ⊕ O(−1)_{E_{∞}}. Then (g_{c}, ω_{c}) is the Kähler quotient at momentum level z_{0} + z_{∞} = 2 of the metric

∑_{a} \frac{(1 + x_{a})z_{0} + (1 − x_{a})z_{∞}}{2x_{a}} g_{a} + \frac{dz_{0}^{2}}{2z_{0}} + \frac{dz_{∞}^{2}}{2z_{∞}} + 2z_{0}θ_{0}^{2} + 2z_{∞}θ_{∞}^{2},

2On each fibre of p̂: M̂ → Ŝ, η̂/4π integrates to 1 and so [η̂/4π] restricts to give the generator of H^{2}(p̂^{−1}(x), ℤ). Hence by the Leray–Hirsch theorem, H^{2}(M̂, ℝ) is generated by [η̂] and pullbacks from S. The restriction of [η̂/2π] to ê_{0} is the first Chern class [ωŜ/2π] of L̂ (and the restriction to ê_{∞} is the first Chern class [−ωŜ/2π] of L̂^{−1}). Thus [η̂/4π] is a projective version of the Thom class of a vector bundle.

where x_{0} = 1, x_{∞} = −1, and θ_{0}, θ_{∞} are connection 1-forms for the U(1)-line bundles O(−1)_{E_{0}}, O(−1)_{E_{∞}} with θ_{0}(K_{0}) = 1 = θ_{∞}(K_{∞}), dθ_{0} = −ω_{0}+Ω_{0}, dθ_{∞} = ω_{∞}+Ω_{∞}.

If we regard M^{0} as an open subset of M̂ = P (O ⊕ L̂), then the diagonal action is generated by K_{0} +K_{∞}, θ_{∞} −θ_{0} is basic and so induces a unitary connection θ (with respect to the quotient U(1)-action) on L̂ with dθ = ωŜ. Substituting z_{0} = 1 + z and z_{∞} = 1 − z and performing the quotient yields (1) with Θ = Θ_{c}. (On each fibre over Ŝ this is the realization of ℂP^{1} as a Kähler quotient of ℂ^{2}.) □

Remark 2. The existence of the canonical metric on M shows there does exist a cohomology class Ξ whose pullback to M̂ is Ξ̂. Ξ is then unique, and the admissible Kähler classes form a family of dimension #A + 1. If b_{2}(S_{a}) = 1 for all a and b_{1}(S_{a}) ≠ 0 for at most one a, then every Kähler class on M is admissible.

1.4. Admissible metrics. Let M = P (E_{0} ⊕ E_{∞}) → S be an admissible bundle and Ω an admissible Kähler class corresponding to parameters x_{a}. Then a Kähler metric in Ω is said to be admissible if it has the form (1) on M^{0}, up to scale, with respect to the given projective unitary bundle structure on M and local Kähler product structure on S. According to Theorem 3, in order for a (scale of a) metric of the form (1) on M^{0} to define an admissible Kähler metric on M, it is necessary that Θ is a smooth function on [−1, 1] satisfying (2)–(3). We now show that these conditions are also sufficient and provide a parameterization of admissible metrics.

We first note that any metric of the form (1), where Θ is a smooth function on [−1, 1] satisfying (2)–(3), defines a smooth metric g on M compatible with the same symplectic form as the canonical Kähler metric g_{c} in Ω, provided that we take z to be the momentum map and θ the connection 1-form of the canonical Kähler metric; then, using (3), we find that g − g_{c} is smooth on M, and g is positive definite on M since it is on M^{0} by (2) and ω is nondegenerate on M. (See [5, §1] for details.)

With this point of view, the smooth functions Θ on [−1, 1] satisfying (2)–(3) define a family of complex structures on M. However, we claim that there is an S^{1}-equivariant biholomorphism, in the identity component of the diffeomorphism group, between any two such complex structures, so that Θ parameterizes Kähler metrics compatible with the given (fixed) complex structure on M whose Kähler forms belong to a given admissible Kähler class Ω. This claim holds essentially because it is true for toric complex structures on ℂP^{1} (and for toric varieties in general), but for later use we need to make explicit the transformation of M that relates the complex and symplectic points of view, following [29, 28, 15].

A key ingredient in this transformation is the notion of a symplectic potential of an admissible Kähler metric defined by Θ(z), which is a function u(z) on (−1, 1) with u′′(z) = U(z) := 1/Θ(z). Then

u_{c}(z) = \frac{1}{2}((1 − z) log(1 − z) + (1 + z) log(1 + z) − 2 log 2)

is the unique symplectic potential for the canonical Kähler metric (g_{c}, J_{c}) given by Θ_{c}(z) = (1 − z)(1 + z), which satisfies u_{c}(±1) = 0. We can extend this description to all admissible Kähler metrics compatible with ω, thanks to the following lemma, which is an easy application of l’Hôpital’s Rule and Taylor’s Theorem.

Lemma 2. A smooth function Θ(z) = 1/U(z) satisfies (3) if and only if U(z) − U_{c}(z) is smooth on [−1, 1]. Then U(z)/U_{c}(z) is positive and smooth on [−1, 1].

On M^{0}, the symplectic potential u(z) of an admissible Kähler metric is closely related to a Kähler potential of ω with respect to J by a fibrewise Legendre transform (see [4, 29]) over Ŝ. Indeed, if we put

y = u′(z), h(y) = −u(z) + yz,

then d_{J}^{c}y = θ and dd_{J}^{c}h(y) = ω − ∑_{a} ω_{a}/x_{a} on (M^{0}, J).^{3} Let y_{c}, h_{c}(y_{c}) denote the corresponding quantities associated to u_{c}. There are local 1-forms α on Ŝ such that θ = dt + α, where t: M^{0} → ℝ/2πℤ is locally defined up to an additive constant on each fibre. Since exp(y + it) and exp(y_{c} + it) give ℂ^{×}-coordinates on the fibres, there is a U(1)-equivariant fibre-preserving diffeomorphism Ψ of M^{0} over Ŝ with

Ψ^{∗}y = y_{c}, Ψ^{∗}t = t, and hence Ψ^{∗}J = J_{c}.

As J_{c} and J are integrable complex structures, Ψ extends to a U(1)-equivariant diffeomorphism of M leaving fixed any point on e_{0}∪e_{∞} (since it is fibre preserving).

Put ω̃ := Ψ^{∗}ω. Then ω̃ is a Kähler form on (M, J_{c}) which (we claim) belongs to the same cohomology class Ω as ω. Indeed, on M^{0} we have

ω̃ − ω = dd_{J_{c}}^{c}(h(y_{c}) − h_{c}(y_{c}))

since dd_{J_{c}}^{c}h(y_{c}) = Ψ^{∗}dd_{J}^{c}h(y) = ω̃ − ∑_{a} ω_{a}/x_{a}, so the following implies the claim.

Lemma 3. The function h(y_{c}) − h_{c}(y_{c}) is smooth on M.

Proof. Since Ψ is a diffeomorphism with Ψ^{∗}y = y_{c}, this holds if and only if h(y) − h_{c}(y) is smooth on M. We already know that h(y) − h_{c}(y_{c}) = −(u(z) − u_{c}(z)) + z(u′(z) − u′_{c}(z)) is smooth (by Lemma 2) so it suffices to show that h_{c}(y) − h_{c}(y_{c}) is smooth on M. However, knowing u_{c} explicitly, we calculate

h_{c}(y) − h_{c}(y_{c}) = −\frac{1}{2}( log (\frac{1 − z̃}{1 − z}) + log (\frac{1 + z̃}{1 + z})),

where z̃ := Ψ^{∗}z is the momentum map of ω̃ = Ψ^{∗}ω; since Ψ is S^{1}-equivariant and fixes e_{0} ∪ e_{∞}, it follows that z̃, viewed as a function of z, satisfies z̃(±1) = ±1; moreover, since both z̃ and z are momentum maps of the same U(1) action on M (and are therefore Morse-Bott functions with the same critical sets), we must have z̃′(±1) ≠ 0. Thus h_{c}(y) − h_{c}(y_{c}) is smooth on M. □

Hence the moduli space K_{ω}^{adm} of admissible metrics in Ω = [ω] is identified with the space of smooth functions Θ on [−1, 1] satisfying (2)–(3) or equivalently with {u ∈ C^{0}([−1, 1]) : u − u_{c} ∈ C^{∞}([−1, 1]), u(±1) = 0 and u′′ > 0 on (−1, 1)}.

1.5. The isometry Lie algebra. For a compact Kähler manifold (M, g), we denote by 𝔦_{0}(M, g) the Lie algebra of all Killing vector fields with zeros. Since M is compact this is equivalently the Lie algebra of all hamiltonian Killing vector fields.

Proposition 2. Let g be an admissible metric on M = P (E_{0} ⊕E_{∞}) →^{p} S and equip S and Ŝ → S with the metrics g_{S}, gŜ induced by ∑_{a} g_{a}/x_{a} on ∏_{a} S_{a}. Let 𝔷(K, g) be the centralizer in 𝔦_{0}(M, g) of the Killing vector field K = J grad_{g} z.

Then the vector space 𝔷(K, g) is the direct sum of a lift of 𝔦_{0}(Ŝ, gŜ) and the span of K in such a way that p_{∗}: 𝔦_{0}(M, g) → 𝔦_{0}(S, g_{S}) is induced by the natural surjection 𝔦_{0}(Ŝ, gŜ) → 𝔦_{0}(S, g_{S}).

Proof. Let X be a holomorphic vector field on Ŝ which is hamiltonian with respect to ω_{h} := ∑_{a} ω_{a}/x_{a}; then the projection X_{a} of X onto the distribution H_{a} (induced by T S_{a} on the universal cover ∏_{a} S_{a} of Ŝ) is a Killing vector field with zeros, so ι_{X_{a}}ω_{h} = −df_{a} for some function f_{a} (with integral zero). Thus ∑_{a} f_{a}x_{a} is a hamiltonian for X with respect to the symplectic form ωŜ = ∑_{a} ω_{a}: since this is the curvature dθ of the connection on M^{0}, X lifts to a holomorphic vector field X̃ = X_{H} + (∑_{a} f_{a}x_{a})K on M^{0}, which is hamiltonian with potential ∑_{a}(1 + x_{a}z)f_{a} and commutes with K. Here X_{H} is the horizontal lift to M^{0} with respect to θ. X̃ and its potential extend to M since M \ M^{0} has codimension ≥ 2 and X̃ has zeros.

3It follows that if ±H_{a} is a local Kähler potential for ±ω_{a} and ũ = u(z) − ∑_{a}(1 + x_{a}z)H_{a}/x_{a}, then ỹ = ∂ũ/∂z is pluriharmonic and h̃ = −ũ + ỹz is a local Kähler potential for ω on (M^{0}, J) [4].

Conversely any element of 𝔷(K, g) pulls back to a holomorphic vector field V on M̂. The projection of V to the normal bundle p̂^{∗}T Ŝ of p̂: M̂ → Ŝ is holomorphic hence constant on the ℂP^{1} fibres by Liouville’s Theorem (the normal bundle is trivial on each fibre), so V is projectable; since V is the pullback of a Killing vector field which commutes with K, it maps to zero iff it comes from a constant multiple of K. This gives a projection to 𝔦_{0}(Ŝ, gŜ) splitting the inclusion just defined. □

2. Admissible extremal Kähler metrics

2.1. Automorphisms and the Matsushima–Lichnerowicz obstruction. On any compact Kähler manifold (M, g), the Lie algebra 𝔥(M) of holomorphic vector fields lies in an exact sequence:

0 → 𝔥_{0}(M) → 𝔥(M) → H^{1}(M, ℝ)^{∗}

where 𝔥_{0}(M) is the ideal of holomorphic vector fields with zeros, which is the Lie algebra of the reduced automorphism group H_{0}(M) ⊂ Aut_{0}(M), i.e., the connected component of the kernel of the Albanese map Aut_{0}(M) → H^{1}(M, ℝ)^{∗}/H_{1}(M, ℤ). The Matsushima–Lichnerowicz Theorem [36, 41] says that if g is CSC, 𝔥_{0}(M) is the complexification of the Lie algebra 𝔦_{0}(M, g) of hamiltonian Killing vector fields and 𝔥(M) = 𝔞(M) ⊕ 𝔥_{0}(M), 𝔞(M) being the central subalgebra of parallel vector fields: thus 𝔥(M) is reductive. This condition on 𝔥(M) is often called the Matsushima– Lichnerowicz obstruction to the existence of CSC Kähler metrics on (M, J).

Let us consider the special case of geometrically ruled complex manifolds M = P (E), where p: P (E) → S and E is a holomorphic vector bundle of rank r + 1 over a compact Kähler 2d-manifold (S, h, ω_{h}). Thus M is a kählerian 2m-manifold with m = r + d: for instance, [O(1)_{E}] + k[p^{∗}ω_{h}] is a Kähler class for k ≫ 1, where O(−1)_{E} is the (fibrewise) tautological line bundle of P (E).

The projection of any holomorphic vector field V ∈ 𝔥(M) to the normal bundle p^{∗}T S is constant on each fibre, so V descends to a holomorphic vector field p_{∗}V ∈ 𝔥(S). Since p_{∗}[V_{1}, V_{2}] = [p_{∗}V_{1}, p_{∗}V_{2}], we have an exact sequence of Lie algebras

0 → 𝔥_{S}(M) → 𝔥_{0}(M) → 𝔥_{0}(S),

where 𝔥_{S}(M) is the subspace of 𝔥(M) of holomorphic vector fields tangent to the fibres of p (which have zeros). Obviously 𝔥_{S}(M) = H^{0}(S, 𝔰𝔩(E)) is the Lie algebra of holomorphic vector fields preserving the ℂP^{r}-fibres of p: P (E) → S. Since an ideal in a reductive Lie algebra is reductive, we obtain the following weaker (but often more useful) version of the Matsushima–Lichnerowicz obstruction.

Proposition 3. Let M = P (E) → S be a geometrically ruled complex manifold which admits a CSC Kähler metric. Then 𝔥_{S}(M) must be reductive.

The following elementary result yields a simple application of this criterion.

Proposition 4. Let M = P (E_{0} ⊕ E_{1} ⊕ ⋯ ⊕ E_{ℓ}), where E_{j} are holomorphic vector bundles on a compact kählerian manifold S such that H^{0}(S, 𝔤𝔩(E_{j})) is reductive and H^{0}(S, Hom(E_{j}, E_{i})) = 0 for all i < j. Then 𝔥_{S}(M) is reductive iff H^{0}(S, Hom(E_{i}, E_{j})) = 0 for all i < j.

Proof. Any element of the Lie algebra 𝔥_{S}(M) = H^{0}(S, 𝔰𝔩(E)) may be represented as an (ℓ+1)×(ℓ+1) matrix (a_{ij}) with a_{ij} ∈ H^{0}(S, Hom(E_{i}, E_{j})). By assumption, this matrix is upper-triangular. The strictly upper-triangular matrices form a nilpotent ideal 𝔫 and if this is zero, 𝔥_{S}(M) is clearly reductive. Conversely, taking commutators with elements of the form Id_{E_{i}}/ rank E_{i} − Id_{E_{j}}/ rank E_{j} ∈ H^{0}(S, 𝔰𝔩(E)), we see that 𝔫 ⊆ [𝔥_{S}(M), 𝔥_{S}(M)]. Hence if 𝔫 ≠ 0, [𝔥_{S}(M), 𝔥_{S}(M)] is not semisimple, i.e., 𝔥_{S}(M) is not reductive. □

Corollary 2. Let M = P (E) where E = L_{0} ⊕ L_{1} ⊕ ⋯ ⊕ L_{r} is a direct sum of holomorphic line bundles over a Riemann surface Σ of genus g. If g ≥ 2 and deg L_{i} −deg L_{j} > g−1 for some 0 ≤ i, j ≤ r then M admits no CSC Kähler metric. If g ≤ 1, then M admits a CSC Kähler metric if and only if deg L_{i} = deg L_{j} for all i, j (i.e., L_{0} ⊕ L_{1} ⊕ ⋯ ⊕ L_{r} is a polystable vector bundle).

Proof. We can assume without loss that E = E_{0} ⊗ ℂ^{r_{0}} ⊕ E_{1} ⊗ ℂ^{r_{1}} ⊕ ⋯ ⊕ E_{ℓ} ⊗ ℂ^{r_{ℓ}} with deg E_{i} ≤ deg E_{j} and E_{i} ≅ E̸_{j} for i < j. The Kodaira vanishing theorem then implies H^{0}(Σ, E_{j}^{−1} ⊗ E_{i}) = 0 for any i < j, and we may apply Proposition 4. By assumption, deg E_{ℓ} − deg E_{0} > max(0, g − 1), and so dim H^{0}(Σ, E_{0}^{−1} ⊗ E_{ℓ}) > 0 by Riemann–Roch. Hence 𝔥_{Σ}(M) is not reductive, and there is no CSC Kähler metric on M. The converse when g ≤ 1 follows from Narasimhan–Seshadri [43]. □

Remark 3. The assumptions of Proposition 4 hold if E_{1}, … E_{ℓ} are projectivelyflat hermitian vector bundles such that the slopes μ(E_{j}) := \overline{c}_{1}(E_{j}) ∪ Ω^{d−1}, with respect to some Kähler class Ω on S (dim S = 2d), satisfy μ(E_{i}) < μ(E_{j}) for i < j. Indeed in this case 𝔤𝔩(E_{j}) is a flat hermitian bundle and H^{0}(S, 𝔤𝔩(E_{j})) is the space of parallel sections of 𝔤𝔩(E_{j})^{4}, which is a complexification of the space of parallel sections of 𝔲(E_{j}), hence a reductive Lie algebra. The slope condition then ensures H^{0}(S, Hom(E_{j}, E_{i})) = 0 for all i < j by a theorem of Kobayashi [33].

In general, the condition that 𝔥_{S}(M) is reductive does not imply 𝔥_{0}(M) is. However it does if p_{∗}: 𝔥_{0}(M) → 𝔥_{0}(S) is surjective and 𝔥_{0}(S) is reductive. This obviously holds if 𝔥_{0}(S) = 0. It also holds if (S, g_{S}) is CSC and there is a metric g on M such that p_{∗} is a surjection from 𝔦_{0}(M, g) to 𝔦_{0}(S, g_{S}). This is true for admissible bundles by Proposition 2.

Proposition 5. Let Ω be an admissible Kähler class on M = P (E_{0} ⊕ E_{∞}) → S, where the local product metric g_{S} on S is CSC. Then the admissible metrics in Ω are invariant under a common maximal compact connected subgroup of H_{0}(M), and 𝔥_{0}(M) is reductive iff H^{0}(S, Hom(E_{0}, E_{∞})) = 0 = H^{0}(S, Hom(E_{∞}, E_{0})). This latter condition holds if \overline{c}_{1}(E_{∞}) − \overline{c}_{1}(E_{0}) is strictly indefinite.

Proof. Let g be an admissible Kähler metric on M. We know by the Matsushima– Lichnerowicz theorem that 𝔥(S) is reductive and 𝔥_{0}(S) is the complexification of the Lie algebra 𝔦_{0}(S, g_{S}) of a maximal compact subgroup of H_{0}(S). Then by Proposition 2, both p_{∗}: 𝔥_{0}(M) → 𝔥_{0}(S) and p_{∗}: 𝔦_{0}(M, g) → 𝔦_{0}(S, g_{S}) are surjective.

To show that the Lie algebra 𝔦_{0}(M, g) is a maximal compact subalgebra of 𝔥_{0}(M), it therefore suffices to show that 𝔦_{0}(M, g)∩𝔥_{S}(M) is a maximal compact subalgebra of 𝔥_{S}(M). Since \overline{c}_{1}(E_{∞})−\overline{c}_{1}(E_{0}) = ω_{S}, we can certainly arrange that μ(E_{∞})−μ(E_{0}) is nonzero by the choice of a Kähler class on S. Then, by Remark 3, we are under the hypotheses of Proposition 4, and, as in its proof, we have that 𝔥_{S}(M) is the direct sum of the reductive centralizer of K and a nilpotent ideal 𝔫 in [𝔥_{S}(M), 𝔥_{S}(M)]. The first part now follows easily from Proposition 2 and Remark 3.

As noted above, 𝔥_{0}(M) is reductive iff 𝔥_{S}(M) is. By Proposition 4, the latter happens iff both H^{0}(S, Hom(E_{0}, E_{∞})) = 0 = H^{0}(S, Hom(E_{∞}, E_{0})). This indeed holds if \overline{c}_{1}(E_{∞})−\overline{c}_{1}(E_{0}) is strictly indefinite (by the vanishing theorem of Kobayashi [33] as in Remark 3), since we can then choose Kähler classes on S such that the corresponding slopes have μ(E_{∞}) − μ(E_{0}) with either sign. □

4This is a standard Bochner argument, as in [33]. Alternatively, note that the pullback of 𝔤𝔩(E_{j})

to the universal cover of S is trivialized by parallel sections, and apply the open mapping theorem:

the pullback of a holomorphic section of 𝔤𝔩(E_{j}) has closed bounded image in this trivialization.

Since S is a local Kähler product, it is CSC iff the factors S_{a} (a ∈ A) in the universal cover are CSC.

2.2. The Futaki invariant and extremal vector field. On a compact Kähler 2m-manifold (M, J, g, ω), recall that the (normalized) Futaki invariant of a real holomorphic vector field with zeros V = J grad_{g} f + grad_{g} h is defined by

𝔉_{ω}(V) = (∫_{M} μ_{g} ∫_{M}(f + ih)Scal_{g}μ_{g} − ∫_{M} Scal_{g}μ_{g} ∫_{M}(f + ih)μ_{g})/Vol(M)^{2},

where μ_{g} = ω^{m}/m! is the volume form of g. Futaki [21] showed that this complex number is independent of the choice of metric in the Kähler class Ω = [ω], and that the map 𝔉_{Ω} : 𝔥_{0}(M) → ℂ is a character on 𝔥_{0}(M). 𝔉_{Ω} is closely related to the Futaki–Mabuchi extremal vector field K_{Ω} := J grad_{g} pr_{g}Scal_{g} of (M, J, Ω, G), where G is a maximal compact connected subgroup of H_{0}(M) and pr_{g} is the L_{2}-projection onto the space of Killing potentials with respect to any G-invariant metric g in Ω: Futaki and Mabuchi [22] showed that K_{Ω} is independent of this choice. Clearly 𝔉_{Ω} and K_{Ω} vanish if Ω contains a CSC metric. Calabi [8] showed that if 𝔉_{Ω} vanishes then any extremal Kähler metric in Ω is a CSC metric, but the vanishing of 𝔉_{Ω} does not suffice in general for the existence of a CSC metric in Ω.

Let Ω be an admissible Kähler class on M = P (E_{0} ⊕ E_{∞}) → S and suppose in addition that for a ∈ A, ±g_{a} is a CSC Kähler metric with scalar curvature Scal_{±g_{a}} = ±2d_{a}s_{a}. Let p_{c}(t) = ∏_{a}(1 + x_{a}t)^{d_{a}} and define

\begin{aligned}α_{r} & := ∫_{−1}^{1} p_{c}(t)t^{r}dt, \\ β_{r} & := p_{c}(1) + (−1)^{r}p_{c}(−1) + ∫_{−1}^{1}(∑_{a} \frac{d_{a}s_{a}x_{a}}{1 + x_{a}t})p_{c}(t)t^{r}dt. \end{aligned} ; (7)

We now compute the Futaki invariant 𝔉_{Ω}(K) of K = J grad_{g} z and show that K_{Ω} is essentially 𝔉_{Ω}(K)K, where G the maximal compact connected subgroup of H_{0}(M) of Proposition 5 preserving admissible Kähler metrics in Ω. 𝔉_{Ω}(K) will reappear in the next paragraph as the leading coefficient of a polynomial associated with Ω.

Proposition 6. Suppose M is admissible over a CSC base and Ω is an admissible Kähler class with admissible metric g. Then 𝔉_{Ω}(K) = 2(α_{0}β_{1} − α_{1}β_{0})/α_{0}^{2}. Also the L_{2}-projection of Scal_{g} orthogonal to the space of Killing potentials is

(8)  Scal_{g} + Az + B

where A and B are given by

\begin{aligned}Aα_{1} + Bα_{0} & = −2β_{0} \\ Aα_{2} + Bα_{1} & = −2β_{1}. \end{aligned} ; (9)

(Since α_{0}α_{2} > α_{1}^{2}, this system has a unique solution for A, B.) In particular the extremal vector field of (Ω, G) is K_{Ω} = −AK = 2(α_{0}β_{1} − α_{1}β_{0})K/(α_{0}α_{2} − α_{1}^{2}).

Proof. We may rescale Ω so that an admissible metric (g, ω) in Ω is of the form (1). We then have

μ_{g} = \frac{ω^{m}}{m!} = p_{c}(z)(⋀_{a} \frac{(ω_{a}/x_{a})^{d_{a}}}{d_{a}!} ) ∧ dz ∧ θ,

where m = 1 + ∑_{a} d_{a} is the complex dimension of M. Thus

∫_{M} μ_{g} = 2πVol(S, ∏_{a} \frac{ω_{a}}{x_{a}} )α_{0} (= Vol(M)),
∫_{M} zμ_{g} = 2πVol(S, ∏_{a} \frac{ω_{a}}{x_{a}} )α_{1} = Vol(M)α_{1}/α_{0},

where Vol(S, ∏_{a} \frac{ω_{a}}{x_{a}} ) = ∏_{a} Vol(S_{a}, \frac{ω_{a}}{x_{a}} ) in the case when S is a global product. The scalar curvature of (M, g) is given by

(10)  Scal_{g} = ∑_{a} \frac{2d_{a}s_{a}x_{a}}{1 + x_{a}z} − \frac{F′′(z)}{p_{c}(z)}

where F (z) = Θ(z)p_{c}(z) (see e.g. [4, (79)]). We thus calculate

∫MzScal_{g}μ_{g} = 2πVol(S, ∏_{a} \frac{ω_{a}}{x_{a}} ) ∫_{−1}^{1}(( ∑_{a} \frac{2d_{a}s_{a}x_{a}}{1 + x_{a}z} )p_{c}(z) − F′′(z))zdz

= 2πVol(S, ∏_{a} \frac{ω_{a}}{x_{a}} )(∫_{−1}^{1}(∑_{a} \frac{2d_{a}s_{a}x_{a}}{1 + x_{a}z} )p_{c}(z)z dz − [zF′(z) − F (z)]_{−1}^{+1})

= 2Vol(M)β_{1}/α_{0},

where we integrate by parts, then impose the boundary conditions (3). Similarly,

∫_{M} Scal_{g}μ_{g} = 2Vol(M)β_{0}/α_{0}

and the first claim follows.

For the second claim note that the above integral formulae imply Scal_{g} + Az + B is orthogonal to the Killing potentials 1, z if and only if (9) holds. By the form of Scal_{g}, the fact that the s_{a} are constant, and Proposition 2, the result follows. □

Note that the above expression for 𝔉_{Ω}(K) is manifestly independent of the choice of a smooth function Θ(z) satisfying (3), as it should be according to the general theory [21]. Indeed, as we have already discussed in §1.4, these smooth functions Θ(z) define Kähler metrics within the same Kähler class.

2.3. K-energy and the extremal polynomial. Given a complex 2m-manifold (M, J), a maximal compact connected subgroup G of H_{0}(M), and a Kähler class Ω, we denote by M_{Ω} the infinite dimensional Fréchet space of Kähler metrics in Ω and let M_{Ω}^{G} be the subspace of G-invariant Kähler metrics in Ω. Following Guan [28] and Simanca [48], consider the map

g ↦ pr_{g}^{⊥}Scal_{g}μ_{g},

where pr_{g}^{⊥} is the L_{2}-projection orthogonal to the space of Killing potentials. This can be viewed (by integration) as a 1-form σ on M_{Ω}^{G}, which turns out to be closed. Therefore for any ω_{0} ∈ Ω, there exists a unique functional E_{ω_{0}}^{G}: M_{Ω}^{G} → ℝ with

dE_{ω_{0}}^{G} = −σ,

E_{ω_{0}}^{G}(ω_{0}) = 0. Note that changing the base point ω_{0} ∈ M_{Ω} would change E_{ω_{0}}^{G} by an additive constant. We refer to E_{ω_{0}}^{G} as the (modified) K-energy: it agrees with the Mabuchi K-energy [38] when G is trivial.

By definition, it is clear that the critical points of E_{ω_{0}}^{G} are exactly the extremal Kähler metrics in M_{Ω}^{G}, since σ = 0 means that Scal_{g} is a Killing potential. Note that by the Calabi Theorem [7], any extremal Kähler metric g ∈ M_{Ω} belongs to M_{Ω}^{G} with G = Isom_{0}(M, g) ∩ H_{0}(M).

Building on earlier work by Bando–Mabuchi [6], Chen [9], Donaldson [16] and others, Chen and Tian have established the following uniqueness result and necessary condition for existence of an extremal Kähler metric.

Theorem 4. [10, 11] Extremal Kähler metrics in M_{Ω} are unique up to automorphism and any extremal Kähler metric in M_{Ω}^{G} realizes the absolute minimum of E_{ω_{0}}^{G} (for any ω_{0} ∈ M_{Ω}^{G}). In particular, if M_{Ω}^{G} contains an extremal Kähler metric, then E_{ω_{0}}^{G} is bounded from below.

Now let M be an admissible projective bundle over a CSC base as in the previous paragraph. We want to obtain a formula for the K-energy as a functional acting on K_{ω}^{adm}, where ω is fixed, so we need to use the description given in §1.4 which shows how K_{ω}^{adm} is embedded into M_{Ω}^{G}, in which the complex structure is fixed.

This description shows that if u_{t}(z) is a path of symplectic potentials in K_{ω}^{adm}, then the smooth functions h_{t}(y_{c}) − h_{c}(y_{c}) define a path ω + dd_{J_{c}}^{c}(h_{t}(y_{c}) − h_{c}(y_{c})) in M_{Ω}, where h_{t} are introduced by

h_{t}(y_{t}(z)) = −u_{t}(z) + y_{t}z, y_{t} = u′_{t}(z),

so that, after setting z = y_{t}^{−1}(y_{c}), we have

h_{t}(y_{c}) = −u_{t}(y_{t}^{−1}(y_{c})) + y_{t}^{−1}(y_{c})y_{c}.

Differentiating with respect to t, we get for the corresponding vector fields u̇ ∈ T_{g}(K_{ω}^{adm}) and ḣ ∈ T_{ω}(M_{Ω}^{G}), the relation (cf. [28, 15]):

ḣ = −u̇.

Hence we obtain the following symplectic version of the (modified) K-energy.

Lemma 4. The K-energy E_{ω}^{G}, restricted to the space of admissible Kähler metrics in Ω and viewed as a function on the space of symplectic potentials, is determined uniquely up to an additive constant by the formula

(dE_{ω}^{G})_{g}[u̇] = ∫_{M}(pr_{g}^{⊥}Scal_{g})u̇ μ_{g},

where pr_{g}^{⊥} denotes the L_{2}-projection orthogonal to the space of Killing potentials.

Consider an admissible metric g in Ω corresponding to the function Θ(z) = F (z)/p_{c}(z). Since the base S is CSC we have, by Proposition 6, pr_{g}^{⊥}Scal_{g} = Scal_{g} + Az + B, with A and B given by (9) and Scal_{g} by (10).

Lemma 5. There is a unique smooth function F_{Ω} on [−1, 1] with

(11)  F′′_{Ω}(z) = (Az + B + ∑_{a} \frac{2d_{a}s_{a}x_{a}}{1 + x_{a}z} )p_{c}(z)

and F_{Ω}(±1) = 0. F_{Ω} satisfies (6) and is a polynomial of degree ≤ m + 2, the coefficient of z^{m+2} being a nonzero multiple of A.

Proof. There is clearly a unique solution to (11) with F_{Ω}(±1) = 0. One easily checks, using (9) that the solution is

(1 + z)p_{c}(−1) + (1 − z)p_{c}(1) + ∫_{−1}^{1}(\frac{1}{2}(At + B) + ∑_{a} \frac{d_{a}s_{a}x_{a}}{1 + x_{a}t})p_{c}(t)|z − t|dt.

The derivative of this function is

p_{c}(−1) − p_{c}(1) + ∫_{−1}^{1}(\frac{1}{2}(At + B) + ∑_{a} \frac{d_{a}s_{a}x_{a}}{1 + x_{a}t})p_{c}(t)sign(z − t)dt,

which gives the formulae for F′_{Ω}(±1) in (6), using the first equation of (9). □

The motivation for this lemma is that now Scal_{g}+Az+B = (F′′_{Ω}(z)−F′′(z))/p_{c}(z), see (10). Furthermore, F and F_{Ω} satisfy the same boundary conditions (6).

Proposition 7. Let Ω be an admissible Kähler class on an admissible bundle over a CSC base. Then the K-energy restricted to the space of admissible Kähler metrics K_{ω}^{adm} is (up to an additive constant) a positive multiple of the functional

E_{g_{c}} : u(z) ↦ ∫_{−1}^{1} F_{Ω}(z)(u′′(z) − u′′_{c}(z))dz − ∫_{−1}^{1} p_{c}(z) log (\frac{u′′(z)}{u′′_{c}(z)})dz.

Proof. E_{g_{c}} is well-defined by Lemma 2 and its gradient is

(dE_{g_{c}})_{g}[u̇] = ∫_{−1}^{1} F_{Ω}(z)u̇′′(z)dz − ∫_{−1}^{1} p_{c}(z)\frac{u̇′′(z)}{u′′(z)}dz = ∫_{−1}^{1}(F_{Ω}(z) − F (z))u̇′′(z)dz.

Integrating twice by parts, using the fact that F and F_{Ω} both satisfy (6), and multiplying by 2πVol(S, ∏_{a} ω_{a}/x_{a}), we obtain ∫_{M}(Scal_{g} + Az + B)u̇μ_{g}. □

Remark 4. It is worth noticing that (by (3)) the functional

E: u(z) ↦ ∫_{−1}^{1}(F_{Ω}(z)u′′(z) − p_{c}(z) log u′′(z))dz

is well-defined on K_{ω}^{adm} and is still a primitive of the restriction of the 1-form σ to K_{ω}^{adm}. It can then be regarded as a natural determination of a (modified) Kenergy on K_{ω}^{adm}, with no explicit reference to any base point in K_{ω}^{adm}, and such that E_{g_{c}}(u) = E(u) − E(u_{c}).

Corollary 3. If there is an extremal Kähler metric in Ω, then F_{Ω} ≥ 0 on [−1, 1].

Proof. If there is an extremal Kähler metric in Ω, then by Theorem 4 [10, 11], the K-energy is bounded from below. We now apply an argument from [15]: take any nonnegative smooth function f(z) with supp(f) ⊂ (−1, 1) and consider the sequence u_{k}(z) with u′′_{k}(z) = u′′_{c}(z) + kf(z) of symplectic potentials (cf. Lemma 2) for admissible Kähler metrics. We therefore get

E_{g_{c}}(u_{k}) = − ∫_{−1}^{1} p_{c}(z) log(1 + k \frac{f(z)}{u′′_{c}(z)})dz + k ∫_{−1}^{1} F_{Ω}(z)f(z)dz.

This will tend to −∞ if ∫_{−1}^{1} F_{Ω}(z)f(z)dz < 0 for some f. □

In the next paragraph, where we complete the proof of Theorem 1, we shall show that positivity of F_{Ω} on (−1, 1) is a necessary and sufficient condition for the existence of an extremal Kähler metric in Ω.

Definition 1. Let Ω be an admissible Kähler class on M. Then the polynomial F_{Ω} constructed above will be called the extremal polynomial of Ω.

2.4. A characterization of extremal admissible Kähler classes. In this paragraph we prove Theorem 1 in three steps. First, if the extremal polynomial F_{Ω} of an admissible Kähler class Ω is positive on (−1, 1), we construct an admissible extremal Kähler metric in Ω by adapting an argument of Guan and Hwang (cf. [27, 30, 31]): we discuss their work further in the next section. Second, we extend the continuity argument of [54] to prove the existence of admissible extremal Kähler metrics for Ω sufficiently small. Third, we use Corollary 3, the uniqueness result of Chen– Tian [10, 11] and an argument from [54] to show that an extremal Kähler metric in Ω is admissible up to automorphism. Hence we deduce that the existence of an extremal Kähler metric in Ω implies that F_{Ω} is positive on (−1, 1).

We begin with the construction. By Proposition 1, an admissible metric (1) is extremal exactly when for each a ∈ Â, ±g_{a} is a CSC Kähler metric with Scal_{±g_{a}} = ±2d_{a}s_{a} and (4)–(5) hold for a polynomial P of degree ≤ N + 1, where N = #Â. The metric g is CSC iff P has degree ≤ N.

We have seen that the boundary conditions (3) imply (6) and the converse clearly holds if p_{c}(±1) ≠ 0 (i.e., d_{0} = 0 = d_{∞}). However, if g is extremal, then (4)–(5) imply that F′′(z) = p′_{c}(z)Υ(z) with Υ(−1) = 2(d_{0} +1) if d_{0} > 0 and Υ(1) = −2(d_{∞} +1) if d_{∞} > 0 (because of the normalization of the Fubini–Study metrics on S_{0} and S_{∞}). Hence, by (6), F′(z) = p_{c}(z)Ψ(z) with Ψ(−1) = 2(d_{0} + 1) and Ψ(1) = −2(d_{∞} + 1), and Θ(±1) = 0. Now by l’Hôpital’s Rule, Θ′(±1) = ∓2. Hence, for extremal Kähler metrics, the boundary conditions (3) are equivalent to (6).

In summary, to obtain a globally defined admissible extremal metric on a projective bundle P (E_{0} ⊕ E_{∞}) → S, we need, for CSC Kähler metrics (±g_{a}, ±ω_{a}) satisfying \overline{c}_{1}(E_{∞}) − \overline{c}_{1}(E_{0}) = ∑_{a}[ω_{a}/2π], to solve (4) and (5) for a polynomial F (of degree ≤ m + 2) which satisfies (6) and is positive on (−1, 1).

For an admissible Kähler class Ω on M, we claim that (4)–(5) and the boundary conditions (6) have a unique solution for F, given by the extremal polynomial F_{Ω}.

Proposition 8. Let M = P (E_{0} ⊕ E_{∞}) → S be an admissible 2m-manifold, where S is CSC. Then for any admissible Kähler class Ω on M, the extremal polynomial F_{Ω} is the unique polynomial F of degree ≤ m + 2 satisfying (4)–(5) and (6).

Proof. An admissible Kähler class on M is specified by parameters x_{a} such that x_{0} = 1, x_{∞} = −1 and otherwise 0 < |x_{a}| < 1 with ω_{a}/x_{a} positive. We write Scal_{±g_{a}} = ±2d_{a}s_{a}. Equation (5) can be solved for a degree N − 1 polynomial P_{0} by Lagrange interpolation, i.e.,

P_{0}(z) = ∑_{a} 2d_{a}s_{a}x_{a} ∏_{b∈Â}(1 + x_{b}z)

and then we can write the general degree N + 1 solution as

P (z) = P_{0}(z) + (Az + B) ∏_{a∈Â}(1 + x_{a}z) = (Az + B + ∑_{a} \frac{2d_{a}s_{a}x_{a}}{1 + x_{a}z} ) ∏_{a∈Â}(1 + x_{a}z)

so that

F′′(z) = (Az + B + ∑_{a} \frac{2d_{a}s_{a}x_{a}}{1 + x_{a}z} )p_{c}(z).

Integrating F′′(z) and zF′′(z) on [−1, 1], (6) now implies that A, B satisfy (9). Hence F = F_{Ω} is the unique solution. □

Remark 5. An alternative approach is to solve the initial value problem (at z = −1) for F (z). The boundary conditions at z = 1 then show that A, B satisfy (9). This gives another formula for the extremal polynomial:

(12) F_{Ω}(z) = 2(1 + z)p_{c}(−1) + ∫_{−1}^{z}(At + B + ∑_{a} \frac{2d_{a}s_{a}x_{a}}{1 + x_{a}t} )p_{c}(t)(z − t)dt,

where A and B are given (as usual) by (9).

Proposition 8 shows that the existence of an admissible extremal Kähler metric in Ω is equivalent to the positivity of the extremal polynomial F_{Ω} on (−1, 1). Since the leading coefficient is a nonzero multiple of A, Proposition 6 shows that such a metric will be CSC iff the Futaki invariant 𝔉_{Ω}(K) vanishes.

Remark 6. Since F_{Ω} depends continuously (in fact analytically) on the admissible Kähler class and has roots at ±1 with fixed multiplicities d_{∞} + 1 and d_{0} + 1, it is positive on (−1, 1) for an open subset of such classes. This observation fits in with the general stability result of LeBrun and Simanca [35].

We now show that F_{Ω} is positive on (−1, 1) for sufficiently small Ω.

Proposition 9. Let M = P (E_{0} ⊕ E_{∞}) → S be admissible, where S is a local Kähler product of CSC metrics (±g_{a}, ±ω_{a}). Then there is a nonempty open subset of admissible Kähler classes on M which contain an (admissible) extremal Kähler metric of positive scalar curvature. The admissible Kähler classes containing a CSC metric form a real analytic hypersurface which is nonempty if \overline{c}_{1}(E_{∞}) − \overline{c}_{1}(E_{0}) is strictly indefinite over S (i.e., the definite forms ω_{a} do not all have the same sign).

Proof. As we noted in Remark 6, the extremal polynomial F_{Ω} is positive on (−1, 1) for an open subset of admissible Kähler classes. It remains to see that this open subset is nonempty and to find the CSC metrics in the family. For this, we study the behaviour of F_{Ω} near x_{a} = 0 for all a ∈ A.

Lemma 6. The coefficients A and B defined by (9), as functions of x_{a} (a ∈ A) for |x_{a}| small are given by

(13)  A = −2(2 + d_{0} + d_{∞})∑_{a∈A}d_{a}x_{a} + O(x^{2})
(14)  B = −(1 + d_{0} + d_{∞})(2 + d_{0} + d_{∞})
− 2∑_{a∈A}d_{a}s_{a}x_{a} + 2(d_{0} − d_{∞})∑_{a∈A}d_{a}x_{a} + O(x^{2})

where O(x^{2}) is shorthand for ∑_{a,b∈A} O(x_{a}x_{b}).

The proof is given in Appendix B. It follows that in the limit x_{a} → 0 for all a ∈ A (which does not give a Kähler class), F′′_{Ω}(z)/(1 + z)^{d_{0}}(1 − z)^{d_{∞}} is given by

(15)  −(1 + d_{0} + d_{∞})(2 + d_{0} + d_{∞}) + \frac{2d_{0}(d_{0} + 1)}{1 + z} + \frac{2d_{∞}(d_{∞} + 1)}{1 − z}.

If d_{0} = 0 and d_{∞} = 0, this is negative on (−1, 1) and F_{Ω} is convex. By (6), for some ε > 0, F_{Ω} is positive and increasing on (−1, −1 + ε) and concave if d_{0} > 0, while it is positive and decreasing on (1 − ε, 1) and concave if d_{∞} > ∞. By (15), for sufficiently small Ω, F_{Ω} does not have enough inflection points to have a zero on (−1, 1) and so it is positive there. Hence the set of admissible Kähler classes containing an admissible extremal Kähler metric is nonempty. Since z ∈ [−1, 1], we see that for x_{a} sufficiently small, the scalar curvature −Az − B of g is positive.

Now the Futaki invariant F_{Ω}(K) ∼ A is a rational function of x_{a}, a ∈ A, so the CSC metrics form a real analytic hypersurface. It is then clear from (13) that if x_{a} occur with both signs, A has nonconstant sign for small x_{a}. □

(Using the Matsushima–Lichnerowicz criterion, the CSC existence result in this proposition provides an alternative proof of the last part of Proposition 5.)

Proposition 10. Let M be an admissible projective bundle over a CSC base. Then if an admissible Kähler class on M contains an extremal Kähler metric, this extremal Kähler metric is admissible up to automorphism.

Proof. Consider the set U of admissible Kähler classes that contain an extremal Kähler metric invariant under the maximal compact subgroup G of H_{0}(M) defined in Proposition 5. By LeBrun–Simanca [35], U is open in the set of all admissible Kähler classes. Suppose there is some admissible class Ω_{0} (with parameters x_{a}^{0}) which contains an inadmissible extremal metric g_{0}; by the Calabi Theorem, we can assume that g_{0} is G-invariant, i.e., Ω_{0} ∈ U. By LeBrun–Simanca [35], this implies that in all Kähler classes sufficiently close to Ω_{0}, there are G-invariant extremal Kähler metrics close to g_{0} (in suitable Sobolev spaces; by the Sobolev embedding theorem this also holds in the C^{ℓ}(M) topology, for any ℓ > 0). We have two cases:

• there is an open neighbourhood of Ω_{0} in U for which the extremal polynomial is not positive on (−1, 1);

• there are admissible extremal Kähler metrics in classes arbitrarily close to Ω_{0}.

The first case contradicts the existence of admissible extremal Kähler metrics on M for sufficiently small Ω, i.e., the positivity of the extremal polynomial. Indeed, for such Ω, F_{Ω}(z) has at most two inflection points in (−1, 1) by (15), and an easy case by case analysis (according to whether d_{0}, d_{∞} are zero or positive) then shows that Q_{Ω}(z) := F_{Ω}(z)/(1 + z)^{d_{0}}(1 − z)^{d_{∞}}, as a polynomial in z, has simple roots. However, by assumption, for all Ω in some open neighbourhood of Ω_{0}, the extremal polynomial F_{Ω} is nonnegative (by Corollary 3) but not positive on (−1, 1), so Q_{Ω}(z) has zero discriminant. Since it is analytic in Ω, it is identically zero, a contradiction.

In the second case we apply instead the uniqueness result [10, 11] for extremal Kähler metrics, as in [54]. Let Ω_{k} be a sequence of admissible Kähler classes (with parameters x_{a}^{k}) which converges to Ω_{0} (i.e., x_{a}^{k} converges to x_{a}^{0} for all a) and such that Ω_{k} contains an admissible extremal Kähler metric g̃_{k} which, without loss, is not CSC. By LeBrun–Simanca [35], it follows that for k ≫ 1 there are G-invariant extremal Kähler metrics g_{k} ∈ Ω_{k} which converge to g_{0} in the C^{2}(M) topology. By Chen–Tian [10, 11], g_{k} is the pullback of g̃_{k} by an automorphism Ψ_{k} of (M, J, Ω_{k}, G). We now claim that g_{0} is the pullback by an automorphism of an admissible extremal Kähler metric in Ω_{0}, completing the proof.

To prove the claim, we use the theory of hamiltonian 2-forms of order 1 from [4, 5]. Since g̃_{k} admits such a 2-form with S^{1} action generated by K, so does g_{k} (by Proposition 6, K is a nonzero multiple of the extremal vector field of (Ω_{k}, G) and so is preserved by Ψ_{k}). Now if (g, ω) is any Kähler metric on M for which K = J grad_{g} z generates an isometric S^{1} action, then it follows from [5] that this action comes from a hamiltonian 2-form if and only if it is rigid (meaning that g(K, K) depends only on z) and semisimple (meaning that for any regular value z_{0} of z, the z-derivative at z = z_{0} of the family of Kähler quotient metrics gŜ(z) on the complex quotient Ŝ is parallel and diagonalizable with respect to gŜ(z_{0})). Thus, the S^{1} action generated by K is rigid and semisimple with respect to g_{k}, hence also with respect to g_{0} by continuity, so that g_{0} itself admits a hamiltonian 2-form of order 1 with S^{1} action generated by K.

We now apply Theorem 3 and Proposition 1 to g_{0}: it follows that g_{0} is adapted to the bundle structure of M = P (E_{0} ⊕ E_{∞}) → S and induces a CSC Kähler metric gŜ^{0} on Ŝ = P (E_{0}) ×_{S} P (E_{∞}), and a connection 1-form θ_{0} on the principal ℂ^{×}- bundle over Ŝ (whose total space is identified with M^{0}), such that the (1, 1)-form dθ_{0} =: ωŜ^{0} is parallel and diagonalizable with respect to gŜ^{0}. Then (by Chern– Weil theory) we have [ωŜ^{0}] = [ωŜ], where ωŜ is the product CSC metric defining the admissible class Ω_{0}. It follows that the Kähler form of gŜ^{0} is in the cohomology class ∑_{a}[ω_{a}]/x_{a}^{0}, since, as explained in §1.3, this cohomology class is determined by the admissible class Ω_{0}, which can be uniquely written as the sum of the ‘projective Thom class’ Ξ̂, of M̂ = P (O ⊕ L̂) → Ŝ and a pullback from Ŝ.

Now by Chen–Tian [10, 11] again (see also [14, 40]), there is an automorphism ψ of Ŝ with gŜ = ψ^{∗}gŜ^{0}, since these are CSC Kähler metrics in the same Kähler class. In fact, the proof of [10, 11] essentially shows that any two extremal metrics in a given Kähler class can be connected by a geodesic in the space of Kähler potentials, and therefore ψ can be chosen in the reduced (connected) automorphism group H_{0}(Ŝ) ⊆ Aut_{0}(Ŝ) (see e.g. [28, 26]); in particular, such a ψ acts trivially on cohomology. By Proposition 2 there is a fibre-preserving S^{1}-equivariant automorphism Ψ of M̂ = P (O ⊕ L̂) → Ŝ, which induces ψ on Ŝ. Thus, Ψ preserves the ℂ^{×}-bundle structure of M^{0} → Ŝ and sends the connection 1-form θ_{0} to a connection 1-form θ̃_{0} with curvature dθ̃_{0} = ψ^{∗}ωŜ^{0} ∈ [ωŜ^{0}] = [ωŜ]; now since ψ^{∗}ωŜ^{0} and ωŜ are both parallel (and therefore harmonic) with respect to ψ^{∗}gŜ^{0} = gŜ, Hodge theory implies they are equal. We can therefore send θ̃_{0} to θ via a bundle isomorphism.

Thus we have constructed (on M^{0} and hence, by a standard extension argument, everywhere) an automorphism sending g_{0} to an admissible extremal Kähler metric in Ω_{0}, as required. □

Theorem 1 follows from Propositions 8–10.

3. Existence and nonexistence results for extremal Kähler metrics

In this section we use Theorem 1 to construct explicit examples of extremal Kähler metrics. We also obtain some nonexistence results for CSC Kähler metrics.

3.1. Constructing admissible extremal Kähler metrics. We begin with a root counting argument due to Hwang [30] and Guan [27] which gives a complete construction when the base S is a local Kähler product of nonnegative CSC Kähler metrics (in fact Hwang and Guan only considered the case that S has constant nonnegative eigenvalues of the Ricci tensor, but the proof is no different in general, and the idea to weaken this hypothesis is already explored in [31]).

Proposition 11. Suppose that M = P (E_{0} ⊕ E_{∞}) → S is admissible where S is a local Kähler product of nonnegative CSC metrics. Then every admissible Kähler class contains an (admissible) extremal Kähler metric.

Proof. By the boundary conditions F_{Ω} is positive, and increasing or decreasing, on (−1, −1 + ε) or (1 − ε, 1) respectively, for some ε > 0. Suppose it is not positive on (−1, 1). Then it has at least two maxima, one minimum and two inflection points on (−1, 1). It follows that P has at least two roots in (−1, 1).

Let y_{1} ≤ ⋯ ≤ y_{Q} and z_{1} ≤ ⋯ ≤ z_{R} (Q, R ≥ 0) denote the roots (counted with multiplicity) of P in [1, ∞) and (−∞, −1] respectively, and put y_{0} = 1, y_{Q+1} = ∞, z_{0} = −∞, z_{R+1} = −1. We order {x_{a} : a ∈ Â} as

−1 ≤ x_{a_{1}} < ⋯ < x_{a_{J}} < 0 < x_{a_{J+1}} < ⋯ < x_{a_{N}} ≤ 1

(for some 0 ≤ J ≤ N) so that g_{a_{j}} is negative definite (hence with s_{a_{j}} nonpositive) for j ≤ J and positive definite (hence with s_{a_{j}} nonnegative) for j ≥ J + 1.

Therefore by (5), for each 0 ≤ q ≤ Q, there is at most one x_{a_{j}} with y_{q} ≤ −1/x_{a_{j}} < y_{q+1}, so that Q + 1 ≥ J with equality iff there is exactly one x_{a_{j}} in each such interval. Similarly, for each 0 ≤ r ≤ R, there is at most one x_{a_{j}} with z_{r} < −1/x_{a_{j}} ≤ z_{r+1}, so that R + 1 ≥ N − J with equality iff there is exactly one x_{a_{j}} in each such interval. Thus P has at least N − 2 roots outside (−1, 1).

Since P has degree ≤ N + 1, it has at most N − 1 roots outside (−1, 1), so we must either have Q + 1 = J or R + 1 = N − J. If (without loss of generality) Q + 1 = J then −1/x_{a_{1}} < y_{1}, so that P (−1/x_{a_{1}}) > 0 (by (5) again) and there must be a root of P between −1/x_{a_{1}} (1 ≤ −1/x_{a_{1}} < y_{1}) and the last maximum of F_{Ω} in (−1, 1). This now forces R + 1 = N − J also, hence P (−1/x_{a_{N}}) > 0 and there must be a root of P between the first maximum of F_{Ω} in (−1, 1) and −1/x_{a_{N}} (z_{R} < −1/x_{a_{N}} ≤ −1), contradicting deg P ≤ N + 1. □

Because of this result, in the rest of this section we shall mainly be interested in the influence of negative scalar curvature factors in the base metrics. In the presence of such factors, the existence of extremal Kähler metrics is nontrivial, as was already observed in [54] for ruled surfaces. By Theorem 1, such a metric exists in a given admissible class Ω iff the extremal polynomial F_{Ω} is positive on (−1, 1). However, the integrals α_{i} and β_{j} involved in the above construction of F_{Ω} are hard to compute in general (see Appendix B). For the next examples, we therefore adopt a different approach to compute F_{Ω}. Instead of solving (4)–(5) and integrating, we solve first the boundary conditions.

It is easy to see that (6), together with (5) for a = 0 and a = ∞ (if there are blow-downs), are solved by any F of the form

(16)  F (z) = (1 − z^{2})(p_{c}(z) + (1 + z)^{d_{0}+1}(1 − z)^{d_{∞}+1}q(z))

for some polynomial q(z). Conversely any polynomial solution is of this form, and to obtain the extremal polynomial F_{Ω}, the degree of q must be ≤ (∑_{a∈A} d_{a}) − 1. Now it remains to compute F′′(z), to solve (4)–(5) for a ∈ A (so that F = F_{Ω}), and to check positivity. For a given projective bundle and admissible Kähler class this leads to equations on the coefficients of q. The Futaki invariant will be zero (and the metric will be CSC) iff q has degree ≤ (∑_{a∈A} d_{a}) − 2.

In general, the algebraic equations on q are hopelessly complicated. However, when S has real dimension ≤ 4, they are tractable.

3.2. Extremal Kähler metrics over a Riemann surface. We consider first extremal Kähler metrics on projective bundles over a Riemann surface, generalizing the study of projective line bundles in [54].

Let Σ be a compact Riemann surface with CSC metric (±g_{Σ}, ±ω_{Σ}) and let M = P (E_{0} ⊕ E_{∞}) → Σ, where E_{0}, E_{∞} are projectively-flat hermitian vector bundles with ranks d_{0} + 1 > 0, d_{∞} + 1 > 0, and \overline{c}_{1}(E_{∞}) − \overline{c}_{1}(E_{0}) = [ω_{Σ}/2π]. Let ±2s be the scalar curvature of ±g_{Σ} and Ω be an admissible Kähler class on M defined by 0 < |x| < 1. The metric

g = (z + 1)g_{0} + (z + 1/x)g_{Σ} + (z − 1)g_{∞} + \frac{p_{c}(z)}{F (z)}dz^{2} + \frac{F (z)}{p_{c}(z)}θ^{2}

is an extremal metric in the given class iff F = F_{Ω}. From §3.1, we know that

F_{Ω}(z) = (1 + z)^{d_{0}+1}(1 − z)^{d_{∞}+1}((1 + xz) + c(1 − z^{2})),

where q(z) = c is a constant uniquely determined by the equation

F′′_{Ω}(−1/x) = 2sx(1 − 1/x)^{d_{0}}(1 + 1/x)^{d_{∞}}.

(This holds whether or not d_{0}, d_{∞} are zero.) We solve this to obtain

c(s, x) = −\frac{2x^{2}(2+d_{0}(1+x)+d_{∞}(1−x)−sx)}{(2+d_{0}(1+x)+d_{∞}(1−x))(4+d_{0}(1+x)+d_{∞}(1−x))+(4+d_{0}+d_{∞})(1−x^{2})}.

Since sx has same sign as Scal_{Σ}, it can be positive only when Σ = ℂP^{1}, in which case E_{0} = L_{0} ⊗ ℂ^{d_{0}+1} and E_{∞} = L_{∞} ⊗ ℂ^{d_{∞}+1} for some line bundles L_{0}, L_{∞}. It then follows that ω_{Σ} is integral, and thus s = p/q where p ≤ 2 and q is an integer of same sign as x ∈ (−1, 1) \ {0} (see Remark 1); then we have that sx < 2, so that c < 0. Therefore 𝔉_{Ω}(K), which is a nonzero multiple of c, doesn’t vanish for any admissible Kähler class. Since b_{2}(Σ) = 1, every Kähler class on M is admissible, so we get an immediate nonexistence result.

Theorem 5. Let E_{0}, E_{∞} be projectively-flat hermitian vector bundles over a Riemann surface Σ. Then there are no CSC Kähler metrics on M = P (E_{0} ⊕ E_{∞}) unless \overline{c}_{1}(E_{0}) = \overline{c}_{1}(E_{∞}) (i.e., E_{0} ⊕ E_{∞} is polystable).

This partially extends the converse in Corollary 2 to the case g > 1. Compared to Theorem 1, we note that here \overline{c}_{1}(E_{∞}) − \overline{c}_{1}(E_{0}) can never be strictly indefinite. On the other hand, by Theorem 1, we have an extremal Kähler metric for sufficiently small Ω. Indeed it is easy to see that |c| is small when |x| is small, and hence (1 + xz) + c(1 − z^{2}) is positive on (−1, 1). We also know from Proposition 11 that if Σ has genus 0 or 1, then every admissible Kähler class contains an extremal Kähler metric. Let us now see what happens when g > 1, i.e., when sx < 0.

Since c < 0, the quadratic Q(z) = (1 + xz) + c(1 − z^{2}) is concave. It is clearly positive at z = ±1, so it is positive on (−1, 1) unless its minimum is in (−1, 1) and it is nonpositive there. The minimum value 1 + c + x^{2}/4c occurs at z = x/2c and

c(s, 1) = −\frac{2(1 + d_{0}) − s}{2(1 + d_{0})(2 + d_{0})}, c(s, −1) = −\frac{2(1 + d_{∞}) + s}{2(1 + d_{∞})(2 + d_{∞})}.

It follows that if s < −d_{0}(d_{0} + 1), then c(s, 1) < −\frac{1}{2} and hence for 0 < x < 1 sufficiently close to 1, we have c(s, x) < −\frac{1}{2} and the minimum of Q(z) is in (−1, 0) and nonpositive. Similarly, if s > d_{∞}(d_{∞} + 1) then for −1 < x < 0 sufficiently close to −1, we have c(s, x) < −\frac{1}{2} and the minimum of Q(z) is in (0, 1) and nonpositive.

Hence if Σ has genus g > 1 and s < −d_{0}(d_{0} + 1) or s > d_{∞}(d_{∞} + 1) then not every admissible Kähler class contains an admissible extremal metric.

Conversely if s ≥ −d_{0}(d_{0} + 1) then it is easy to check that c(s, x) ≥ −x/2 for all 0 < x < 1, so the minimum of Q(z) is not in (−1, 1) for any such x, whereas if s ≤ d_{∞}(d_{∞} + 1), c(s, x) ≥ x/2 for all −1 < x < 0 and again the minimum of Q(z) is not in (−1, 1) for any such x.

Theorem 6. Let E_{0}, E_{∞} be projectively-flat hermitian vector bundles of ranks d_{0}+1, d_{∞}+1 over a compact Riemann surface Σ of genus g, and suppose \overline{c}_{1}(E_{∞})− \overline{c}_{1}(E_{0}) = [ω_{Σ}/2π] for a Kähler form ±ω_{Σ} of constant curvature. Then there exist admissible extremal Kähler metrics on P (E_{0} ⊕ E_{∞}) → Σ. Such metrics exist in every Kähler class if g = 0 or 1. For g > 1, put ρ_{Σ} = sω_{Σ}. Then such metrics exist in every Kähler class if and only if −d_{0}(d_{0} + 1) ≤ s ≤ d_{∞}(d_{∞} + 1), otherwise such metrics exist for |x| sufficiently small (depending on s).

Remark 7. In absence of blow-downs, we recover the examples of [54] on (complex) pseudo-Hirzebruch surfaces P (O⊕L) → Σ, where there are Kähler classes which do not contain an extremal Kähler metric (if Σ has genus g > 1). Our result extends these examples to higher rank projective bundles. However, in the presence of blow-downs, there do exist projective bundles for which there is an extremal Kähler metric in every Kähler class, even with g > 1.

3.3. Nonexistence of CSC Kähler metrics over a Hodge 4-manifold. We now obtain a similar nonexistence result to Theorem 5 when dim S = 4.

Theorem 7. Let (S, ±g_{S}, ±ω_{S}) be a CSC Hodge 4-manifold and let E_{0}, E_{∞} be projectively-flat hermitian vector bundles of ranks d_{0} + 1, d_{∞} + 1 over S with \overline{c}_{1}(E_{∞}) − \overline{c}_{1}(E_{0}) = [ω_{S}/2π]. Then there are no CSC Kähler metrics in the admissible Kähler classes on M = P (E_{0} ⊕ E_{∞}) → S.

Proof. We will prove that 𝔉_{Ω}(K) is nonzero for any admissible Kähler class Ω by showing that the leading coefficient of F_{Ω} cannot vanish. Following the discussion in §3.1 we see that for given d_{0}, d_{∞} ≥ 0 and 0 < |x| < 1 we have

F_{Ω}(z) = (1 + z)^{d_{0}+1}(1 − z)^{d_{∞}+1}((1 + xz)^{2} + (cz + e)(1 − z^{2}))

with c and e being constants uniquely determined by the conditions

F′′_{Ω}(−1/x) = 0  and  \frac{F′′_{Ω}(z)}{1 + xz} |_{z=−1/x} = 4sx(1 − 1/x)^{d_{0}}(1 + 1/x)^{d_{∞}}.

The leading coefficient of F_{Ω} vanishes iff c = 0. As before c (and e) are determined by s and x. In particular, c(s, x) = n(s, x)/d(s, x), where

d(s, x) = (1 + d_{∞})(2 + d_{∞})^{2}(3 + d_{∞})(1 − x)^{4}
+ 4(2 + d_{0})(1 + d_{∞})(2 + d_{∞})(3 + d_{∞})(1 − x)^{3}(1 + x)

+ 6(2 + d_{0})(2 + d_{∞})(4 + d_{0} + d_{∞} + (1 + d_{0})(1 + d_{∞}))(1 − x)^{2}(1 + x)^{2}

+ 4(1 + d_{0})(2 + d_{0})(3 + d_{0})(2 + d_{∞})(1 − x)(1 + x)^{3}
+ (1 + d_{0})(2 + d_{0})^{2}(3 + d_{0})(1 + x)^{4}

(which is manifestly positive for |x| < 1) and

−n(s, x)/2x^{3} = 4(6 − 3sx + sx^{3})
+ d_{0}(1 + x)((5 − x)(1 + x) + (7 − x)(3 − sx))
+ d_{∞}(1 − x)((5 + x)(1 − x) + (7 + x)(3 − sx))

+ (9 − sx)(d_{0}(1 + x) + d_{∞}(1 − x))^{2} + (d_{0}(1 + x) + d_{∞}(1 − x))^{3}.

Since s = p/q where p ≤ 3 (see Remark 1) and q is an integer of same sign as x ∈ (−1, 1) \ {0}, we have that sx < 3 and a moment’s thought then gives that n(s, x), and therefore c(s, x), is never zero. □

3.4. CSC Kähler metrics over a product of two Riemann surfaces. As counterpoint to the nonexistence results of §§3.2–3.3, we now explore explicitly the existence of CSC Kähler metrics, given by Theorem 1, in the simplest case when the base is a global product of two Riemann surfaces and there are no blow-downs.

Let Σ_{a} (a = 1, 2) be compact Riemann surface with CSC metrics (±g_{a}, ±ω_{a}) and let M be P (O ⊕ L) → Σ_{1} × Σ_{2} where L = L_{1} ⊗ L_{2} and L_{a} are pullbacks of line bundles on Σ_{a} with c_{1}(L_{a}) = [ω_{a}/2π]. Let ±2s_{a} be the scalar curvature of ±g_{a} and −1/x_{a} be the constant roots defining an admissible Kähler class with x_{1} ≠ x_{2} (the case x_{1} = x_{2} was considered in §3.3, where we established nonexistence of CSC metrics). We thus have p_{c}(z) = (1 + x_{1}z)(1 + x_{2}z) and the metric becomes

g = \frac{1 + x_{1}z}{x_{1}} g_{1} + \frac{1 + x_{2}z}{x_{2}} g_{2} + \frac{p_{c}(z)}{F (z)}dz^{2} + \frac{F (z)}{p_{c}(z)}θ^{2}.

According to §3.1, to obtain a CSC metric, F (z) must be the extremal polynomial

F_{Ω}(z) = (1 − z^{2})((1 + x_{1}z)(1 + x_{2}z) + cx_{1}x_{2}(1 − z^{2})),

where c is a constant such that the following relations are satisfied:

F′′_{Ω}(−1/x_{1}) = 2s_{1}(x_{1} − x_{2})  and  F′′_{Ω}(−1/x_{2}) = 2s_{2}(x_{2} − x_{1}).

Writing 2(1 − c) = s (which is \frac{1}{6}Scal_{g} and not to be confused with the s in the previous two paragraphs), these relations hold iff

(17)  x_{1}(s_{1}(x_{1} − x_{2}) − 2 + (1 − s)x_{1}x_{2}) + 3(s − 1)x_{2} = 0
(18)  x_{2}(s_{2}(x_{2} − x_{1}) − 2 + (1 − s)x_{1}x_{2}) + 3(s − 1)x_{1} = 0,

and these are precisely the conditions on an admissible Kähler class Ω (parameterized by (x_{1}, x_{2}) with 0 < |x_{a}| < 1) coming from the vanishing of 𝔉_{Ω}(K) (see Proposition 8). Eliminating s = 2(1 − c), we obtain (using x_{1} ≠ x_{2})

(19)  x_{1}(6 + s_{1}x_{1}(x_{2}^{2} − 3)) + x_{2}(6 + s_{2}x_{2}(x_{1}^{2} − 3)) = 0.

The normalized scalar curvatures s_{a} are subject to the integrality conditions s_{a} = 2(1 − g_{a})/q_{a} for q_{a} a nonzero integer with the same sign as x_{a}, where g_{a} is the genus of Σ_{a}. In particular s_{a}x_{a} < 2. This and (19) imply that x_{1}x_{2} < 0; we thus get a nonexistence result in the case x_{1}x_{2} > 0.

Theorem 8. Let (Σ_{a}, ω_{a}) (a = 1, 2) be compact CSC Riemann surfaces and L be a holomorphic vector bundle over Σ_{1} × Σ_{2} with c_{1}(L) = [(ω_{1} + ω_{2})/2π] (so that c_{1}(L) is positive definite). Then there are no admissible Kähler classes on M = P (O ⊕ L) ≅ P (O ⊕ L^{−1}) containing a CSC Kähler metric.

Remark 8. Note that we do not need to assume that the base is a global product of compact Riemann surfaces for the nonexistence result in the above theorem. It is sufficient to have a compact base S that is a local product of Riemann surfaces with CSC and s_{a}x_{a} < 2, which is always satisfied, since Scal_{±g_{a}} ≤ 4 by the integrality of the pull-back of ±ω_{a} to the universal cover of S.

In contrast to this result, we have the following observation.

Lemma 7. Let Ω be an admissible Kähler class, corresponding to a solution (x_{1}, x_{2}) of (17)–(18) with s ≥ 0. Then Ω admits an admissible CSC Kähler metric with scalar curvature 6s.

Proof. If (19) holds, the extremal polynomial F_{Ω} of an admissible Kähler class gives rise to a globally defined CSC Kähler metric iff F_{Ω} > 0 on (−1, 1). Let Q(z) = F_{Ω}(z)/(1 − z^{2}), and observe that the coefficient of z^{2} in this quadratic is \frac{1}{2}sx_{1}x_{2}. Since Q(±1) > 0, Q will be positive on [−1, 1] if it is convex, i.e., if sx_{1}x_{2} < 0. If s = 0, Q(z) is linear and positive on [−1, 1]. Since x_{1}x_{2} < 0, F_{Ω} is positive on (−1, 1) whenever we have solutions of (17)–(18) with s ≥ 0. □

We now obtain some explicit solutions of (17)–(18). If we take x_{2}(17) − x_{1}(18) and x_{1}(17) − x_{2}(18), we obtain, for x_{1} ≠ x_{2}:

(s_{1} + s_{2})x_{1}x_{2} = 3(s − 1)(x_{1} + x_{2})
2(x_{1} + x_{2}) = s_{1}x_{1}^{2} + s_{2}x_{2}^{2} + (1 − s)x_{1}x_{2}(x_{1} + x_{2}).

These are equivalent to (17)–(18) for x_{1}^{2} ≠ x_{2}^{2}. As x_{1} ≠ x_{2}, x_{1}^{2} = x_{2}^{2} iff x_{1} + x_{2} = 0 and then s_{1} + s_{2} = 0. The following lemma deals with this case.

Lemma 8. If s_{1} +s_{2} = 0, then either x_{1} +x_{2} = 0 and s = (1−x_{1}^{2} +2s_{1}x_{1})/(3−x_{1}^{2}), or, without loss, x_{1} = x_{2} + 1, s = 1, and s_{1} = 2 = −s_{2}. Conversely, these give solutions of (17)–(18).

Proof. Clearly s_{1} + s_{2} = 0 iff s = 1 or x_{1} + x_{2} = 0. The formula for s in the latter case is immediate from (17). Now if s = 1, then without loss of generality s_{1} = −s_{2} is nonnegative and we must have either x_{1} + x_{2} = 0, or s_{1} > 0 and x_{1} = x_{2} + 2/s_{1}. Since 0 < |x_{a}| < 1, this forces x_{1} to be positive, hence s_{1} ≤ 2, so in fact we must have s_{1} = 2 and x_{1} = x_{2} + 1. □

In order to apply Lemma 7, we suppose in the first case above that x_{1}s_{1} ≥ 0: then s > 0 since 1 − x_{1}^{2} > 0 for |x_{1}| < 1. Thus in both cases s_{a}x_{a} ≥ 0 for a = 1, 2 and we obtain CSC Kähler metrics on projective line bundles over T^{2} × T^{2}, T^{2} × ℂP^{1} and ℂP^{1} × ℂP^{1}. In particular any Kähler class on P (O ⊕ O(q, −q)) → ℂP^{1} × ℂP^{1} is admissible, so the above Lemmas and Proposition 11 yield the following conclusions.

Theorem 9. On P (O⊕O(q, −q)) → ℂP^{1}×ℂP^{1} (q ≥ 1), any Kähler class (parameterized, up to scale, by 0 < x_{1} < 1 and −1 < x_{2} < 0) contains a unique admissible extremal Kähler metric. For q > 1 this metric is CSC if and only if x_{1} + x_{2} = 0, while for q = 1 it is CSC if and only if x_{1} + x_{2} = 0 or x_{1} = x_{2} + 1.

When q = 1, the two 1-parameter families of CSC Kähler classes of this theorem intersect at x_{1} = 1/2, x_{2} = −1/2. In fact, the CSC metric in this Kähler class is the Koiso–Sakane Kähler–Einstein metric [34, 47].

We end our study of CSC Kähler metrics on P (O ⊕L) → Σ_{1} ×Σ_{2} by considering the case of zero scalar curvature metrics, which we do not obtain automatically from Theorem 1. If s = 0 then equation (17) defines x_{2} as a function of x_{1}

x_{2} = f_{1}(x_{1}) = x_{1} \frac{2 − s_{1}x_{1}}{x_{1}^{2} − s_{1}x_{1} − 3},

whereas (18) defines x_{1} as a function of x_{2}

x_{1} = f_{2}(x_{2}) = x_{2} \frac{2 − s_{2}x_{2}}{x_{2}^{2} − s_{2}x_{2} − 3}.

Note that f_{1}(0) = f_{2}(0) = 0 and the gradients dx_{2}/dx_{1} of the two graphs at x_{1} = x_{2} = 0 are both negative. By comparing the size of the gradients, one sees that for x_{1} small and positive the graph of f_{1} is above the graph of f_{2}. Note also that the denominator appearing in f_{a}(x_{a}) is negative at x_{a} = 0.

Assume that s_{1} ≤ 0. If f_{1} has no asymptotes for 0 < x_{1} < 1 then f_{1}(1) ≤ −1. Otherwise, for the asymptote x_{1} = v closest to x_{1} = 0 we have lim_{x_{1}→v^{−}} = −∞. Assume moreover that 0 < s_{2}. If f_{2} has no asymptotes for −1 < x_{2} < 0 then f_{2}(−1) > 1. Otherwise, for the asymptote x_{2} = v closest to x_{2} = 0, we have lim_{x_{2}→v^{+}} = +∞. By continuity, the graphs of f_{1} and f_{2} intersect in the open square (0, 1) × (0, −1) and (17)–(18) is solved for some 0 < x_{1} < 1 and −1 < x_{2} < 0.

Theorem 10. Let (Σ_{a}, ±ω_{a}) (a = 1, 2) be compact Riemann surfaces with genus g_{a} and canonical bundles K_{a}, and suppose that the Kähler forms ±ω_{a} are integral with constant curvature. Let L_{a} be line bundles on Σ_{a} with c_{1}(L_{a}) = [ω_{a}/2π] and, if g_{a} ≠ 1, let L_{a} be K_{a}^{q_{a}/2(g_{a}−1)} tensored by a flat line bundle, for an integer q_{a}.

There is then an admissible scalar-flat Kähler metric on P (O⊕L_{1}⊗L_{2}) → Σ_{1}×Σ_{2} in the following cases:

• Σ_{1} = T^{2} and L_{1} is ample, Σ_{2} has genus g_{2} > 1 and q_{2} < 0;

• Σ_{1} and Σ_{2} both have genus g_{a} > 1, q_{1} > 0, and q_{2} < 0.

4. K-stability and admissible extremal Kähler metrics

4.1. Introduction to stability. It has been first suggested by S. T. Yau in [56], then formulated and worked out in several ways, in particular by G. Tian [53] and by S. Donaldson [15], that the existence of a Kähler-Einstein or, more generally, of a CSC or an extremal Kähler metric on a projective complex manifold in the Kähler class determined by some polarization L should be equivalent to some kind of stability for the polarized projective variety (M, L). This conjecture is drawn from a detailed formal picture which makes clear an analogy with the well-established relation between the polystability of vector bundles and the existence of Einstein– Hermitian connections.

At present the most promising candidate for the conjectured stability criterion is ‘K-polystability’, in the form given by Donaldson [15], following Tian [53]: a polarized projective variety (M, L) is K-polystable if any ‘test configuration’ for (M, L) has nonpositive Futaki invariant with equality iff the test configuration is a product. We shall explain this definition shortly. We also discuss an idea of J. Ross and R. Thomas [45, 46], who focus on test configurations arising as ‘deformations to the normal cone’ of subschemes of (M, L), leading to a notion of ‘slope’ Kpolystability analogous to the slope polystability of vector bundles. We explore this analogy further in §4.2.

(Note that some authors use the term K-stable rather than K-polystable, but the latter term agrees better with pre-existing notions of stability.)

4.1.1. Finite dimensional motivation. Let (X, L, Ω) be a polarized Kähler manifold with a hermitian metric on L with curvature −iΩ (thus c_{1}(L) = Ω/2π). Suppose a compact connected group G acts holomorphically on X with momentum map μ: X → 𝔤^{∗} (i.e., d⟨μ, ξ⟩ = −Ω(K_{ξ}, ·), where K_{ξ} is the vector field on X corresponding to ξ ∈ 𝔤, the Lie algebra of G). There is a lift of the action to L generated by K̃_{ξ} + ⟨μ, ξ⟩K for each ξ ∈ 𝔤, where ⟨μ, ξ⟩ is pulled back to L, K̃_{ξ} is the horizontal lift, and K generates the standard U(1) action on L. The action of 𝔤 on X and L extends to an action of the complexification 𝔤^{c} and we assume this integrates to an action of a complex Lie group G^{c}.

By a well-known result of Kempf–Ness and Kirwan, for any x ∈ X, there is a g ∈ G^{c} such that μ(g · x) = 0 iff for any nonzero lift x̃ of x to L^{∗}, the orbit G^{c} · x̃ is closed. Such points x are said to be polystable. If X^{ps} denotes the set of polystable points in X, we then have an equality between X^{ps}/G^{c}, the polystable quotient of X by G^{c}, and the symplectic quotient X//G = μ^{−1}(0)/G.

G^{c} · x̃ is closed iff α(ℂ^{×}) · x̃ is closed for any one parameter subgroup α: ℂ^{×} ↪ G^{c}. This leads to the Hilbert–Mumford criterion for polystability: x is said to be semistable if for any one parameter subgroup α: ℂ^{×} ↪ G^{c}, the linear action of ℂ^{×} on L_{x_{0}}^{∗} has nonpositive weight w_{x_{0}}(α) ≤ 0, where x_{0} = lim_{λ→0} α(λ) · x is the limit point; x is then polystable if it is semistable and w_{x_{0}}(α) = 0 only when x_{0} = x; finally x is stable if it is polystable and has zero dimensional isotropy subgroup.

4.1.2. The infinite dimensional analogue. We apply the finite dimensional picture above formally to an infinite dimensional setting in which X is the space of compatible complex structures on a compact symplectic manifold (M, ω) with H^{1}(M) = 0. The space X has a natural Kähler metric with respect to which the group G of symplectomorphisms of M acts holomorphically with a momentum map μ: X → C_{0}^{∞}(M, ℝ) given by the scalar curvature of the corresponding Kähler metric on M, modified by a constant in order to lie in 𝔤^{∗} ≅ 𝔤 = C_{0}^{∞}(M, ℝ), the functions with total integral zero, which is the Lie algebra of the symplectomorphism group equipped with the L_{2}-inner product. A quick way to see this is to observe that the Mabuchi K-energy (see §2.3) of M is a Kähler potential for the metric on X: the gradient on X of the Mabuchi K-energy is the scalar curvature [26].

There is no group whose Lie algebra is the complexification 𝔤^{c}, but one can still consider the foliation of X given by the vector fields induced by 𝔤^{c}. The complex structures in a given leaf are all biholomorphic by a diffeomorphism in the connected component of the identity, and pulling back the symplectic form ω by these biholomorphisms, we may identify the leaf with the set of all Kähler metrics in a fixed Kähler class, compatible with a fixed complex structure on M. Hence there should be a CSC metric in a given Kähler class iff the momentum map μ vanishes on the corresponding leaf iff the leaf is stable in a suitable sense.

To make precise this infinite dimensional analogue, we formalize what is meant by the orbit of a 1-parameter subgroup in terms of ‘test configurations’ and give a Hilbert–Mumford formulation of stability in terms of the weight of limit points.

4.1.3. Test configurations. Let (M, Ω) be a Hodge manifold, viewed as a polarized projective variety with respect to a line bundle L with c_{1}(L) = Ω/2π.

Definition 2. [15] A test configuration for (M, L) is a polarized scheme (X, E) over ℂ with a ℂ^{×} action α and a flat proper ℂ^{×}-equivariant morphism p: X → ℂ (where ℂ^{×} acts on ℂ by scalar multiplication) such that the fibre (X_{t} = p^{−1}(t), E|_{X_{t}}) is isomorphic to (M, L) for some (hence all) t ≠ 0.

(X_{0}, E|_{X_{0}}) is called the central fibre. Since 0 ∈ ℂ is fixed by the action, (X_{0}, E|_{X_{0}}) inherits a ℂ^{×} action, also denoted by α.

A test configuration is said to be a product configuration if X = M × ℂ and α is given by a ℂ^{×} action on M (and scalar multiplication on ℂ).

Since relevant properties of test configurations are unchanged if we replace E by E^{r} for a positive integer r, we can let E be a ℚ-line bundle in the definition above (i.e., E denotes a ‘formal root’ of a line bundle E^{r} for some positive integer r).

A particularly important class of test configurations are those associated to a subscheme of (M, L), as studied by J. Ross and R. Thomas [45, 46]. We shall state it here for complex submanifolds of (M, L), but the same definition actually makes sense for subschemes.

Definition 3 (Deformation to the normal cone). For a polarized complex manifold (M, L) and a complex submanifold Z the normal bundle is defined by ν_{Z} = T M_{|Z}/T Z (in the more general situation when Z is a subscheme of M, ν_{Z} then denotes the normal cone, defined similarly). The deformation to the normal cone is then defined as the family p : X → ℂ, where X = { \atop M × ℂ} denotes the blowup of M × ℂ along Z × {0} and p the map induced by the natural projection from M × ℂ to ℂ. For t ≠ 0 in ℂ, p^{−1}(t) is then biholomrphic to M, whereas p^{−1}(0) = P ∪ M̂, where P denotes the exceptional divisor in { \atop M × ℂ}, which is biholomorphic to ℙ(O_{Z} ⊕ ν_{Z}), hence is a compactification of the normal cone, and M̂ stands for the blow-up of M along Z; notice that P ∩ M̂ = ℙ(ν_{Z}), the exceptional divisor in M̂.^{5} We equip this with the polarization E_{c} = π^{∗}L⊗O(−cP), where O(P) is the line bundle associated to the exceptional divisor P, π: X → M is induced by the natural projection from M × ℂ to M, and c is a positive rational number such that E_{c} is an ample ℚ-line bundle. This last condition gives an upper bound ε on c, called the Seshadri constant of Z with respect to L.

We let α be the ℂ^{×} action coming from the trivial action on M and multiplication on ℂ. This clearly defines an action on X with a lift to E_{c}. Hence the deformation to the normal cone determines a family of test configurations, parameterized by c ∈ (0, ε) ∩ ℚ.

4.1.4. The Futaki invariant and K-stability. K-stability is defined using a Hilbert– Mumford criterion, i.e., in terms of a ‘weight’ associated to each test configuration. This weight is given by the Futaki invariant of the central fibre; however, since the latter is typically a singular projective variety, we need an algebraic geometric definition of the Futaki invariant. Such a definition has been given by Donaldson [15].

5The name deformation to the normal cone originates from the following fact: In the induced holomorphic embedding Z×ℂ ↪ { \atop M × ℂ}, Z×\{t\} ↪ p^{−1}(t) is isomorphic to the natural embedding Z ↪ M, whereas Z × {0} ↪ p^{−1}(0) = P ∪ M̂ has its image in P = ℙ(O_{Z} ⊕ ν_{Z}) and is isomorphic to the natural embedding Z ↪ ν_{Z} ⊂ ℙ(O_{Z} ⊕ ν_{Z}) as the zero section of ν_{Z}, cf. [20, Chapter 5] for more details on this classical construction.

Let V be a scheme of dimension n over ℂ polarized by an ample line bundle L and suppose that α is a ℂ^{×} action on V with a lift to L. Then α acts on the vector spaces H_{k} = H^{0}(V, L^{k}), k ∈ ℤ^{+}. If w_{k}(α) denotes the weight of the highest exterior power of H_{k} (that is, the trace Tr A_{k} of the infinitesimal generator A_{k} of the action) and d_{k} denotes the dimension of H_{k} then w_{k}(α) and d_{k} are given by polynomials in k for sufficiently large k, of degrees at most n+1 and n respectively. For sufficiently large k the quotient w_{k}(α)/(kd_{k}) can be expanded into a power series with no positive powers. The Futaki invariant 𝔉(α) is the residue at k = 0 of this quotient, that is, the coefficient of the k^{−1} term in the resulting expansion. The Futaki invariant is independent of the choice of lift of α to L. (When V is a manifold, this definition coincides with Futaki’s original definition up to a normalization convention.)

Definition 4. The Futaki invariant of a test configuration is defined to be the Futaki invariant 𝔉(α) of the central fibre, where α denotes the induced ℂ^{×} action.

A Hodge manifold (M, L) is said to be K-polystable if the Futaki invariant of any test configuration is nonpositive, and equal to zero if and only if the test configuration is a product configuration.

For the test configurations (X, E_{c}) arising from a deformation to a normal cone, J. Ross and R. Thomas [45, 46] show that the Futaki invariants 𝔉(α_{c}) are rational in c ∈ (0, ε)∩ℚ, where ε is the Seshadri constant, and can be extended to c ∈ (0, ε). With this in mind, we give the following

Definition 5. A Hodge manifold (M, L) is said to be slope K-polystable if for the deformation to the normal cone of any nontrivial subscheme, the Futaki invariant 𝔉(α_{c}) of the corresponding family (X, E_{c}) of test configurations and its natural extension to the whole interval (0, ε) are negative.

Remark 9. The concepts of slope K-stability, slope K-polystability, slope K-semi-stability were introduced by J. Ross and R. Thomas [45], [46] and, in their relative versions, by G. Székelyhidi [50], [51]. The notion of slope K-polystability in [45] is stronger than the one in [46] and has been partially motivated by the limiting situations illustrated by our Example 1 below (cf. Section 2.1 and Remark 3.9 in [45], and Remark 10 below). Our Definition 5 is closer to the one in [45], but is not quite the same as [45] also requires 𝔉(α_{ε}) < 0 unless ε is rational and the semi-ample configuration (X, E_{ε}) is the pullback by a contraction of a product configuration. We shall not need this refinement. As observed in [45, Remark 3.9], the notion of (relative) slope K-polystability used here and in [45] might more properly be referred to as analytic (relative) slope K-polystability, corresponding to analytic test configurations (while the original definitions in [53, 15] only concern algebraic test configurations).

4.2. Stable bundles and CSC Kähler metrics. We now relate our results concerning CSC Kähler metrics on projective bundles to stability theory for vector bundles. Recall that if E → S is a holomorphic vector bundle over a compact kählerian 2d-manifold (S, [ω_{h}]), the slope μ(E) is the number \overline{c}_{1}(E) · [ω_{h}]^{d−1}; E is called (slope) stable or semistable if μ(F) < μ(E) or μ(F) ≤ μ(E) (respectively) for any proper coherent subsheaf F ⊂ E; it is polystable if it is a direct sum of stable vector bundles with the same slope; then, as is well-known, ‘stable’ ⇒ ‘polystable’ ⇒ ‘semistable’, and by the Hitchin–Kobayashi correspondence (established by Donaldson [13] and Uhlenbeck–Yau [55]), E admits an Einstein–Hermitian connection iff it is polystable.

There is a close analogy between K-stability for polarized Kähler manifolds and slope stability for vector bundles. In particular, one might hope to find a direct relation between the existence problem for CSC Kähler metrics on a geometrically ruled complex manifold P (E) over S and the stability of E → S. Notable progress in understanding the relation between K-polystability of P (E) and slope polystability of E has been made by Ross–Thomas [45, 46], using their notion of slope K-polystability: indeed if F is a coherent subsheaf of E, then P (F) is a subscheme of P (E) and deformation to the normal cone of P (F) is a test configaration which ‘destabilizes’ P (E) iff μ(F) > μ(E) (see [45]).

Using the general theory of CSC Kähler metrics, the work of [45] shows that if E is not semistable with respect to an integral Kähler class [ω_{h}] on S, then for all k ≫ 1 the integral classes 2πc_{1}(O(1)_{E}) + kp^{∗}[ω_{h}] on P (E) →^{p} S do not contain CSC metrics. As a partial converse, Hong [32, Theorem A] shows that if E is polystable and 𝔥_{0}(M) → 𝔥_{0}(S) is surjective, then there is a CSC metric in 2πc_{1}(O(1)_{E}) + kp^{∗}[ω_{h}], for each k ≫ 1, iff the Futaki invariant 𝔉_{Ω} vanishes.

To put our results in this context, let P (E) → S be admissible so E = E_{0} ⊕ E_{∞} for projectively-flat (and thus polystable) hermitian vector bundles E_{0} and E_{∞} with \overline{c}_{1}(E_{∞})−\overline{c}_{1}(E_{0}) = ∑_{a}[ω_{a}/2π]. Thus E is determined up to tensor product by a line bundle and is polystable iff it is semistable iff μ(E_{0}) = μ(E_{∞}). With respect to a Kähler class [ω_{h}] = [∑_{a} ω_{a}/f_{a}] on S (where f_{a}x_{a} > 0), this condition reads

(20)  0 = μ(E_{∞}) − μ(E_{0}) = \frac{(d − 1)!}{2π}Vol(S, ∏_{a}ω_{a}/f_{a})(∑_{a}d_{a}f_{a}),

which can happen for suitable f_{a} iff \overline{c}_{1}(E_{∞}) − \overline{c}_{1}(E_{0}) is strictly indefinite; this is exactly the condition of Theorem 1 that ensures the existence of CSC metrics in a sufficiently small admissible Kähler class Ω = Ξ + p^{∗}[∑_{a∈A} ω_{a}/x_{a}], subject only to the constraint that 𝔉_{Ω}(K) = 0. Note that Ξ is equal to 4πc_{1}(O(1)_{E}) up to a basic term (depending on the choice of E)—this follows by integrating Ξ^{d_{0}+d_{∞}+1} over a fibre and using the expression for I(d_{0}, d_{∞}, 0) from Appendix B. Thus, admissible classes play a similar role to those considered by Ross–Thomas and Hong, and k ≫ 1 corresponds to |x_{a}| sufficiently small in our picture. (We recall that this means that the fibres are small compared to the base.) However, there is not a simple relation in general between those Ω containing a CSC metric and the [ω_{h}] with respect to which E is polystable: the approach of Ross–Thomas and Hong suggests taking f_{a} = x_{a}/(1 + r_{a}x_{a}), for some r_{a} depending only on E; then (20) agrees asymptotically with 𝔉_{Ω}(K) = 0 in the limit x_{a} → 0, but the two conditions define distinct hypersurfaces in general.

Conversely, in the case E_{0} and E_{∞} are line bundles over a product S = Σ_{1} × Σ_{2} of two Riemann surfaces, Theorem 8 shows that polystability of E with respect to some Kähler class on S (which is unique up to scale in this case) is also necessary for the existence of a CSC metric in an admissible Kähler class on P (E).

Consider now the case that the base S is a Riemann surface Σ of genus g; the stability of a holomorphic vector bundle is then independent of the choice of a Kähler class on Σ, and it is natural to speculate [45] that the notion of Kpolystability of the projective manifold P (E) should be independent of the specific Kähler class, and to conjecture that P (E) admits a CSC Kähler metric iff E is polystable. At present (see [2]) this conjecture is confirmed when E is of rank 2 (i.e., on geometrically ruled surfaces), when g ≤ 1 and E is a direct sum of line bundles (cf. Corollary 2—this always holds when g = 0), or when E is indecomposable and g ≥ 2. Theorem 5 further confirms the conjecture in the case of decomposable bundles of the form E = E_{0} ⊕ E_{∞} with E_{0} and E_{∞} polystable.

4.3. Extremal Kähler metrics and relative K-polystability. In recent work, G. Székelyhidi [50] has extended the theory of K-polystability to cover extremal Kähler metrics, not just CSC Kähler metrics. We briefly explain his ideas here.

4.3.1. Motivation. Recall that extremal Kähler metrics are critical points for the L_{2}-norm of the scalar curvature for metrics in a fixed Kähler class on a complex manifold (M, J) [7]. If we identify the Kähler class with a leaf of the formal G^{c} orbit described in §4.1.2, we are therefore looking for critical points of ||μ||^{2}, where μ: X → C_{0}^{∞}(M, ℝ) and X is the space of compatible complex structures on a compact symplectic manifold (M, ω) with H^{1}(M) = 0.

We can adapt the finite dimensional model of §4.1.1 to this problem by supposing that the Lie algebra 𝔤 is equipped with a G-invariant inner product ⟨, ⟩. Now, following Székelyhidi [50], we note that the weight w_{x} of the linear action of the isotropy algebra 𝔤_{x} on L_{x}^{∗} is given by w_{x} = ⟨β_{x}, ·⟩: 𝔤_{x} → ℝ for some β_{x} ∈ 𝔤_{x}, which is the orthogonal projection of μ(x) onto 𝔤_{x}. We refer to β_{x} (or rather the induced vector field on X) as the extremal vector field: for in the infinite dimensional setting it agrees with the extremal vector field of Futaki and Mabuchi (see §2.2).

Clearly x is a critical point of ||μ||^{2} iff μ(x) is in 𝔤_{x}, i.e., β_{x} = μ(x). Using this, Székelyhidi shows that x is in the G^{c} orbit of a critical point of ||μ||^{2} if and only if it is polystable for the action of the subgroup of G^{c} whose Lie algebra is the subspace β_{x}^{⊥} of the centralizer of β_{x}. The Hilbert–Mumford criterion may then be modified as follows: the modified weight w_{x_{0}}(α) − ⟨α, β_{x}⟩w_{x_{0}}(β_{x})/⟨β_{x}, β_{x}⟩ of the limit point x_{0} should be nonpositive for any one parameter subgroup α of the centralizer of β_{x}, with equality if and only if x_{0} = x.

4.3.2. The inner product and modified Futaki invariant. Thus motivated, we return to the setting of §4.1.4 and define a modified Futaki invariant of a polarized scheme (V, L) (of dimension n over ℂ) relative to a ℂ^{×} action β. We first need to define an inner product between such actions.

Assume then that V has two ℂ^{×} actions α and β with lifts to L and infinitesimal generators A_{k} and B_{k} of the actions on H_{k}. Then for k sufficiently large, Tr (A_{k}B_{k}) is a polynomial of degree at most n + 2. The inner product ⟨α, β⟩ is defined to be the coefficient of k^{n+2} of the expansion of Tr (A_{k}B_{k}) − w_{k}(α)w_{k}(β)/d_{k} for large k, which is independent of the lifts of α and β to L: indeed it depends only on the trace-free parts of A_{k} and B_{k}. (When V is a manifold, this inner product coincides with Futaki–Mabuchi bilinear form [22] up to a normalization convention.)

We define the modified Futaki invariant [50] 𝔉_{β}(α) of α relative to β (assuming the action β is nontrivial) to be

𝔉_{β}(α) = 𝔉(α) − \frac{⟨α, β⟩}{⟨β, β⟩}𝔉(β).

4.3.3. Relative K-stability. Let (M, Ω, L) be as in §4.1.3 and suppose it has a nontrivial ℂ^{×} action β.

Definition 6. [50] A test configuration (X, E) for (M, L) is compatible with β if there is a ℂ^{×} action, also denoted by β, on (X, E) preserving p: X → ℂ and inducing the trivial action on ℂ, such that β restricted to (X_{t}, E|_{X_{t}}) coincides with the original action for t ≠ 0 under the isomorphism with (M, L).

In this case we have an induced action on the central fibre X_{0}, also called β, and the modified Futaki invariant of the test configuration is defined to be 𝔉_{β}(α).

A polarized Hodge manifold (M, L) with nontrivial ℂ^{×} action β is K-polystable relative to β if the modified Futaki invariant of any test configuration (X, E) compatible with β is nonpositive, and equal to zero if and only if (X, E) is a product.

If (M, L) has a ℂ^{×} action β which preserves a subscheme Z, the test configurations (X, E_{c}) arising from the deformation to the normal cone are compatible with β. As in [45, 46], 𝔉_{β}(α_{c}) is rational in c ∈ (0, ε) ∩ ℚ and so extends to c ∈ (0, ε). Thus, cf. §4.1.4, we have a notion of slope K-polystability relative to β.

Definition 7. A polarized Hodge manifold (M, L) with nontrivial ℂ^{×} action β is said to be slope K-polystable relative to β if the modified Futaki invariant 𝔉_{β}(α_{c}) of the family (X, E_{c}) of test configurations, corresponding to the deformation to the normal cone of any nontrivial subsheme preserved by β, is negative for c ∈ (0, ε).

As with the definition of (absolute) slope K-polystability, strictly speaking, we should also require 𝔉_{β}(α_{ε}) < 0 unless ε is rational and (X, E_{ε}) is the pullback by a contraction of a product configuration.

It follows from [23, 42] that the (Futaki–Mabuchi) extremal vector field associated to a Hodge Kähler manifold (M, Ω) with a maximal compact connected subgroup G of H_{0}(M) has closed orbits, and therefore defines an effective ℂ^{×} action which we will refer to as the extremal ℂ^{×} action of (M, Ω, G). By ‘K-polystable relative to G’, we mean relative to the extremal ℂ^{×} action of (Ω, G). The motivation of §4.3.1 then suggests the following conjecture [50].

Conjecture 1. Let (M, Ω, L) be a polarized Hodge manifold and G a maximal compact connected subgroup of H_{0}(M). Then there is a G-invariant extremal Kähler metric in Ω = 2πc_{1}(L) if and only if (M, L) is K-polystable relative to G.

As a motivating example, Székelyhidi considers the deformation to the normal cone of the infinity section in a (higher genus) polarized geometrically ruled surfaces P (O ⊕ L) → Σ. He finds that relative K-polystability implies existence of extremal Kähler metrics of the type constructed in [54], which are precisely the admissible metrics on these bundles. In the next section we generalize this idea to arbitrary admissible bundles. However, in doing so, we find that unless dim S ≤ 4, we need to replace ‘K-polystable’ by ‘slope K-polystable’ in the above conjecture.

4.4. Relative K-polystability of admissible projective bundles. We now consider the deformation to the normal cone (X, E_{c}, α) of the infinity section e_{∞} = z^{−1}(−1) = P (0 ⊕ E_{∞}) for an admissible projective bundle M = P (E_{0} ⊕ E_{∞}) → S (with dim S = 2d), polarized by a line bundle L with Ω = 2πc_{1}(L) admissible.

We therefore choose the admissible Kähler class Ω = Ξ+∑_{a}[ω_{a}]/x_{a} to be integral (where 0 < |x_{a}| ≤ 1 with equality iff a ∈ {0, ∞}). The Seshadri constant of this polarization is 2, so we take c ∈ (0, 2) ∩ ℚ. Since the ℂ^{×} action β induced by the vector field K preserves Z, X is compatible with this action. We will use the letters α, β to denote also the corresponding actions on the (polarized) central fibre (X_{0}, L_{0}) and on the vector space H^{0}(X_{0}, L_{0}^{k}), where L_{0} = E_{c}|_{X_{0}}.

Let us calculate the modified Futaki invariant of this configuration. For this we first note that if I_{∞} ⊂ O_{M} is the ideal sheaf of holomorphic functions vanishing on e_{∞}, then for any p ≥ 0, I_{∞}^{p}/I_{∞}^{p+1} is supported on e_{∞}, and its restriction is S^{p}ν_{∞}^{∗}, where ν_{∞} is the normal bundle to e_{∞} in M.

Therefore, for k sufficiently large, we have, as in [45, 50]

H^{0}(X_{0}, L_{0}^{k}) = ⨁_{i=0}^{(2−c)k} H^{0}(e_{∞}, L|_{e_{∞}}^{k} ⊗ S^{2k−i}ν_{∞}^{∗}) ⊕ ⨁_{j=1}^{ck} H^{0}(e_{∞}, L|_{e_{∞}}^{k} ⊗ S^{ck−j}ν_{∞}^{∗}),

where α acts on the first direct sum with weight 0 and on the components of the second direct sum with weight −j. We can choose the lift of β to L so that the weight of the induced action on H^{0}(e_{∞}, L|_{e_{∞}}^{k} ⊗ S^{uk+v}ν_{∞}^{∗}) is (u − 1)k + v.

Now S^{p}ν_{∞}^{∗} is the direct image q_{∗}O(p)_{ν_{∞}}, where O(−1)_{ν_{∞}} is the (fibrewise) tautological bundle of q: ê_{∞} = P (ν_{∞}) → e_{∞}. Also ê_{∞} may be identified with Ŝ via the obvious inclusion i of Ŝ into M̂ = P (O ⊕ L̂) as the infinity section, and then i^{∗}O(1)_{ν_{∞}} = L̂. For convenience, we now drop the hats, so that we have

H^{0}(X_{0}, L_{0}^{k}) = ⨁_{i=0}^{(1−z)k} H^{0}(S, i^{∗}L^{k} ⊗ L^{2k−i}) ⊕ ⨁_{j=1}^{(1+z)k} H^{0}(S, i^{∗}L^{k} ⊗ L^{(1+z)k−j})

= ⨁_{i=0}^{2k} H^{0}(S, i^{∗}L^{k} ⊗ L^{2k−i}),

where we have abused notation by writing c−1 = z; a priori this has nothing to do with the momentum map that we also denote by z, but notice that it does take (rational) values in the same interval (−1, 1). To compute d_{k}, Tr A_{k}, Tr B_{k}, Tr A_{k}B_{k}, Tr B_{k}^{2}, and thereby 𝔉_{β}(α), we need only the dimensions of these vector spaces. We note that we only need to compute d_{k}, Tr A_{k} and Tr B_{k} to subleading order in k, whereas for Tr A_{k}B_{k} and Tr B_{k}^{2} the leading order term suffices. Consequently we will be dropping lower order terms without further comment. We also note that since the Futaki invariant is defined in terms of ratios, we can ignore any overall multiples. Now by the Riemann–Roch formula and the ampleness of i^{∗}L (in fact it is only semiample if d_{∞} > 0, but we can apply a limiting argument in this case, as in [45]), for sufficiently large k we have that

h^{0}(S, i^{∗}L^{k} ⊗ L^{uk+v}) = χ(S, i^{∗}L^{k} ⊗ L^{uk+v}) = (ch(i^{∗}L^{k} ⊗ L^{uk+v}) · Td(S))[S]

= (c_{1}(i^{∗}L^{k} ⊗ L^{uk+v}) + \frac{1}{2}c_{1}(K_{S}^{−1}))^{d}[S] + O(k^{d−2})

= (∑_{a} \frac{k + ((u − 1)k + v + s_{a}/2)x_{a}}{x_{a}} [ω_{a}/2π])^{d}[S] + O(k^{d−2})

since the Ricci form of S_{a} may be written s_{a}ω_{a} + ρ_{a,0} where ρ_{a,0} ∧ ω_{a}^{d_{a}−1} = 0. After an overall multiplication by (2π)^{d}/d! and ∏_{a} x_{a}^{d_{a}}/Vol(S_{a}, ω_{a}), this is

k^{d}p_{c}^{s}(u − 1 + v/k) + O(k^{d−2}),

where p_{c}^{s}(t) = ∏_{a}(1 + x_{a}(t + s_{a}/2k))^{d_{a}}. In order to carry out the summations over i and j we use the trapezium rule, as in [45, Lemma 4.7].

Lemma 9. Let f(x) be a polynomial and b a rational number. Then for ε ∈ {0, 1} and for k ∈ ℤ^{+} such that bk is a positive integer, we have

∑_{i=ε}^{bk} f(i/k) = k ∫_{0}^{b} f(t) dt + \frac{1}{2}(f(b) + (−1)^{ε}f(0)) + O(k^{−1}).

The proof is easy (see e.g. [45]): by linearity we can assume f(x) = x^{m} and then use ∑_{i=1}^{N} i^{m} = N^{m+1}/(m + 1) + N^{m}/2 + O(N^{m−1}) (which in turn is an easy induction on N). We then obtain (up to an overall multiple), that for any r ≥ 0,

k^{−d−r}Tr B_{k}^{r} = k ∫_{0}^{2}(1 − t)^{r}p_{c}^{s}(1 − t) dt + \frac{1}{2}(p_{c}(1) + (−1)^{r}p_{c}(−1)) + O(1/k)
= kα_{r} + \frac{1}{2}β_{r} + O(1/k),

34 V. APOSTOLOV, D. CALDERBANK, P. GAUDUCHON, AND C. TØNNESEN-FRIEDMAN

with α_{r} = ∫_{−1}^{1} p_{c}(t)t^{r} dt and β_{r} as in (7). Setting r = 0 gives d_{k}. Similarly, using the explicit formula (12) for the extremal polynomial F_{Ω}(z) we obtain

k^{−d−1}Tr A_{k} = k ∫_{0}^{1+z} −tp_{c}^{s}(z − t) dt − \frac{1}{2}(1 + z)p_{c}(−1) + O(1/k)

= −k ∫_{−1}^{z} p_{c}(t)(z − t) dt − \frac{1}{2} ∫_{−1}^{z}(∑_{a} \frac{d_{a}s_{a}x_{a}}{1 + x_{a}t})p_{c}(t)(z − t)dt

− \frac{1}{2}(1 + z)p_{c}(−1) + O(1/k) = −k ∫_{−1}^{z}(z − t)p_{c}(t) dt

− \frac{1}{4}F_{Ω}(z) + \frac{1}{4} ∫_{−1}^{z}(At + B)p_{c}(t)(z − t) dt + O(1/k)

k^{−d−3}Tr A_{k}B_{k} = ∫_{0}^{1+z} −t(z − t)p_{c}(z − t) dt + O(1/k)

= − ∫_{−1}^{z} p_{c}(t)t(z − t) dt + O(1/k)

where A and B are the solutions of (9). Now we are ready to calculate ⟨β, β⟩, ⟨α, β⟩, 𝔉(β), and 𝔉(α). (We omit the dependence of z for convenience.)

⟨β, β⟩ = \frac{α_{2}α_{0} − α_{1}^{2}}{α_{0}}
⟨α, β⟩ = − ∫_{−1}^{z} p_{c}(t)t(z − t) dt + \frac{α_{1}}{α_{0}} ∫_{−1}^{z} p_{c}(t)(z − t) dt
𝔉(α) = Res_{k=0} \frac{(Tr A_{k})_{1} + (Tr A_{k})_{0}/k}{α_{0}(1 + β_{0}/(2kα_{0}))} = \frac{α_{0}(Tr A_{k})_{0} − \frac{1}{2}β_{0}(Tr A_{k})_{1}}{α_{0}^{2}}

= \frac{−\frac{1}{4}α_{0}F_{Ω}(z) + \frac{1}{4}α_{0} ∫_{−1}^{z}(At + B)p_{c}(t)(z − t) dt + \frac{1}{2}β_{0} ∫_{−1}^{z} p_{c}(t)(z − t) dt}{α_{0}^{2}}

𝔉(β) = Res_{k=0} \frac{α_{1} + β_{1}/2k}{α_{0}(1 + β_{0}/(2kα_{0}))} = \frac{β_{1}α_{0} − β_{0}α_{1}}{2α_{0}^{2}}

where we have set (Tr A_{k})_{0} = −\frac{1}{4}F_{Ω}(z)+ \frac{1}{4} ∫_{−1}^{z}(At+B)p_{c}(t)(z−t) dt and (Tr A_{k})_{1} = −k ∫_{−1}^{z} p_{c}(t)(z − t) dt.

Finally, we can calculate the modified Futaki invariant for our test configuration.

α_{0}^{2}𝔉_{β}(α) = α_{0}^{2}(𝔉(α) − ⟨α, β⟩𝔉(β)/⟨β, β⟩)

= −\frac{1}{4}α_{0}F_{Ω}(z) + \frac{1}{4}α_{0} ∫_{−1}^{z}(At + B)p_{c}(t)(z − t) dt + \frac{1}{2}β_{0} ∫_{−1}^{z} p_{c}(t)(z − t) dt

+ \frac{α_{0}(β_{1}α_{0} − β_{0}α_{1})}{2(α_{2}α_{0} − α_{1}^{2})}(∫_{−1}^{z} p_{c}(t)t(z − t) dt − \frac{α_{1}}{α_{0}} ∫_{−1}^{z} p_{c}(t)(z − t) dt)

= −\frac{1}{4}α_{0}F_{Ω}(z) + \frac{1}{4}α_{0} ∫_{−1}^{z}(At + B)p_{c}(t)(z − t) dt − \frac{1}{4}α_{0} ∫_{−1}^{z} Atp_{c}(t)(z − t) dt

+ \frac{β_{0}(α_{2}α_{0} − α_{1}^{2}) − α_{1}(β_{1}α_{0} − β_{0}α_{1})}{2(α_{2}α_{0} − α_{1}^{2})} ∫_{−1}^{z} p_{c}(t)(z − t) dt

= −\frac{1}{4}α_{0}F_{Ω}(z) + \frac{1}{4}α_{0} ∫_{−1}^{z}(At + B)p_{c}(t)(z − t) dt
− \frac{1}{4}α_{0} ∫_{−1}^{z} Atp_{c}(t)(z − t) dt − \frac{1}{4}α_{0} ∫_{−1}^{z} Bp_{c}(t)(z − t) dt
= −\frac{1}{4}α_{0}F_{Ω}(z)

which is a negative multiple of the extremal polynomial. It follows immediately that if (M, L) is slope K-polystable relative to K = J grad_{g} z, then F_{Ω} is positive on (−1, 1) and Ω contains an admissible extremal metric by Theorem 1.

If (M, L) is slope K-polystable in the absolute sense, then 𝔉(α) is negative on (−1, 1) and hence nonpositive at z = 1. Evaluating the integrals in this case (and using F_{Ω}(1) = 0), we find that A ≥ 0. Now if we swap the roles of the zero and infinity sections (by interchanging E_{0} and E_{∞}) then the analogous calculation shows that A ≤ 0 (we get the same formulae with the change of variables z ↦ −z). Thus A = 0 and 𝔉(β) = 0. (Intuitively, the reason we get 𝔉(β) = 0 is that in these limits, deformation to the normal cone of the zero or infinity section is actually the pullback by a contraction of the product configuration associated to ±β, cf. [45, 46].) Hence (M, L) is slope K-polystable relative to β, F_{Ω} is positive on (−1, 1), and the admissible extremal metric is CSC.

This proves Theorem 2, providing evidence for the reverse implication in Conjecture 1 (with relative K-stability replaced by relative slope K-stability) because in our setting, the extremal vector field is a nonzero multiple of AK. This calculation also shows that the forward implication in Conjecture 1 implies Corollary 3, without referring to K-energy or the results of Chen–Tian [10, 11], providing further indirect evidence. However, if we use relative K-stability instead of relative slope K-stability, we can only deduce that F_{Ω} is positive on (−1, 1) ∩ ℚ and hence nonnegative on (−1, 1). However, since Ω is integral, F_{Ω} has rational coefficients, and so when dim S(= ∑_{a∈A} 2d_{a}) ≤ 4 it follows that F_{Ω} is positive on (−1, 1): indeed F_{Ω}(z) = (1 + z)^{d_{0}+1}(1 − z)^{d_{∞}+1}Q(z) where Q(z) is a quadratic or cubic with rational coefficients, and the repeated roots of such a polynomial must be rational.

On the other hand, the following examples show that positivity of the extremal polynomial on (−1, 1) ∩ ℚ is not sufficient for the existence of an extremal Kähler metric when dim S = 6.

Example 1. Let S = Σ_{1} ×Σ_{2} ×Σ_{3} be a product of hyperbolic Riemann surfaces Σ_{a} with integral Kähler classes [±ω_{a}]. Then for any admissible projective line bundle M over S and any admissible integral Kähler class Ω on M with parameters x_{a} ∈ ℚ, the extremal polynomial has the form

F_{Ω}(z) = (1 − z^{2})(p_{c}(z) + (1 − z^{2})(a_{0} + a_{1}z + a_{2}z^{2})).

where the a_{j} are determined by the constant gaussian curvatures ±s_{a} of Σ_{a} (via (4)– (5)). However, since we are free to choose the genera and degrees of the line bundles defining M, the s_{a} can be arbitrary rational numbers subject only to the constraint that s_{a}x_{a} < 0 (so that the gaussian curvatures are negative). Hence we are free to choose the a_{j} subject to this constraint.

We claim that for any rational r > 0 and x_{1} > x_{2} > 0 > x_{3}, we can choose the a_{j} so that F_{Ω}(z) is a positive multiple of (1 − z^{2})(z^{2} + rz − 1)^{2} provided that

(21)  x_{1}x_{2}x_{3} + x_{1} + x_{2} + x_{3} = 0.

F_{Ω} then has a repeated root in (0, 1) and another in (−∞, −1) and for r in an open subset of ℚ^{+}, these roots are irrational. Obviously for any 1 > x_{1} > x_{2} > 0 rational, (21) has a unique rational solution x_{3} = −(x_{1} + x_{2})/(1 + x_{1}x_{2}) with 0 > x_{3} > −1, and it is elementary to verify our claim by equating coefficients. F′′_{Ω}(z) is then negative for large z and has at least two roots in (−1, 1) and none in (1, ∞). We then check that for r > 8/5, F′′_{Ω}(−1) is negative and so the other two roots (which must be real, since F_{Ω} must have four inflection points) are in (−∞, −1). Hence we can choose x_{1}, x_{2} so that s_{1}x_{1} < 0 and s_{2}x_{2} < 0, with s_{a} defined by (4)–(5). It automatically follows that s_{3}x_{3} < 0.

These data then define a countably infinite family (parameterized by (x_{1}, x_{2}, r) in an open subset of ℚ^{3}) of admissible projective line bundles over products of three Riemann surfaces together with admissible rational Kähler classes (which we can scale to be integral) such that F_{Ω} is positive on (−1, 1) ∩ ℚ, but has an irrational repeated root in (−1, 1). By Theorem 1 these Kähler classes then do not contain an extremal Kähler metric.

Remark 10. According to the conjectures [15, 50], the above examples should define unstable projective varieties. However, the degeneration that demonstrates this instability is not algebraic. While we cannot prove that there is no other algebraic test configuration which would detect this instability, it is difficult to imagine how such a test configuration could be constructed. Thus, presumably, our examples are algebraically (relative) K-polystable but analytically only (relative) K-semistable. This suggests that the non-algebraic degenerations implicit in the use of slope K-polystability may be essential to relate stability to existence of CSC and extremal Kähler metrics.

Appendix A. Relation to previous papers

In this appendix we summarize the classification of compact Kähler 2m-manifolds M with a hamiltonian 2-form of order ℓ given in [5, Theorem 5], and explain how Theorem 3 follows from this classification in the case ℓ = 1. We also give a nonexistence result for extremal Kähler metrics when ℓ = 2.

A.1. Summary of the classification. Let (M, g, J, ω) be a compact connected Kähler 2m-manifold with a hamiltonian 2-form ϕ of order ℓ. Let p(t) be the momentum polynomial of ϕ and K(t) = J grad_{g} p(t) be the corresponding family of hamiltonian Killing vector fields. We summarize results from [4, 5] in italics. The vector fields {K(t) : t ∈ ℝ} generate an effective isometric hamiltonian action of an ℓ-torus 𝕋 on M and p(t) has m − ℓ constant roots counted with multiplicity. This action is free on a connected dense open subset M^{0} of M.

We let S_{Δ} be the stable quotient of M by the induced action of the complexified torus 𝕋^{c} and denote by η_{a}, for a in a finite set with ≤ m − ℓ elements, the distinct constant roots of p(t) and by d_{a} their multiplicities. S_{Δ} is covered by a product S̃_{Δ} = ∏_{a} S_{a} of Kähler 2d_{a}-manifolds (S_{a}, ±g_{a}, ±ω_{a}), and M^{0} → S_{Δ} is a principal 𝕋^{c}-bundle.

In [4, 5], we took a ∈ {1, … N}, but here we shall adopt (in a moment) a different notation for the index set. We let p_{c}(t) = ∏_{a}(t − η_{a})^{d_{a}} and write p(t) = p_{c}(t)p_{nc}(t), where p_{nc}(t) = ∑_{r=0}^{ℓ}(−1)^{r}σ_{r}t^{ℓ−r} and σ_{0} = 1. The Killing vector fields K_{r} := J grad_{g} σ_{r}, for r = 1, … ℓ, are linearly independent on M^{0}.

The image Δ of the momentum map (σ_{1}, … σ_{ℓ}) is a simplex in 𝔱^{∗} ≅ ℝ^{ℓ}, whose interior (the image of M^{0}) is the image under the elementary symmetric functions of a domain D = ∏_{j=1}^{ℓ}(β_{j−1}, β_{j}), where β_{0} < β_{1} < ⋯ < β_{ℓ}. The roots of p_{nc}(t) define smooth, functionally independent, pairwise distinct functions ξ_{j} (j = 1, … ℓ) on M^{0} which extend continuously to M with image [β_{j−1}, β_{j}]. The codimension one faces of Δ may be labelled F_{0}, … F_{ℓ} such that on F_{j}, either ξ_{j} = β_{j} or ξ_{j+1} = β_{j}.

The local description of the metric on M^{0} is as follows (where we have set c = 1, according to [5, Remark 13]). There are 1-forms θ_{1}, … θ_{ℓ} on M^{0} with θ_{r}(K_{s}) = δ_{rs} and dθ_{r} = ∑_{a}(−1)^{r}η_{a}^{ℓ−r}ω_{a} and a function Θ of one variable satisfying

(−1)^{ℓ−j}Θ > 0 on (β_{j−1}, β_{j}), (22)

Θ(β_{j}) = 0, Θ′(β_{j}) = − ∏_{k≠j}(β_{j} − β_{k}), (23)

such that the Kähler structure on M^{0} may be written

g = ∑_{a} p_{nc}(η_{a})g_{a} + ∑_{j=1}^{ℓ} \frac{Δ_{j}}{Θ(ξ_{j})}dξ_{j}^{2} + ∑_{j=1}^{ℓ} \frac{Θ(ξ_{j})}{Δ_{j}}(∑_{r=1}^{ℓ} σ_{r−1}(ξ̂_{j})θ_{r})^{2},

ω = ∑_{a} p_{nc}(η_{a})ω_{a} + ∑_{r=1}^{ℓ} dσ_{r} ∧ θ_{r},

(24)

where ∑_{a} p_{nc}(η_{a})g_{a} is the pullback of a local Kähler product metric on Ŝ, Δ_{j} = ∏_{k≠j}(ξ_{j} − ξ_{k}), and σ_{r}(ξ̂_{j}) is the rth elementary symmetric function of ξ_{1}, … ξ_{ℓ} with ξ_{j} omitted. (σ_{r} itself is the rth elementary symmetric function of ξ_{1}, … ξ_{ℓ}.)

The global description of M in [5, Theorem 5] was presented using the blow-up M̂ of M along the inverse image of the codimension one faces F_{0}, … F_{ℓ} of S_{Δ}. M̂ is 𝕋^{c}-equivariantly biholomorphic to the ℂP^{ℓ}-bundle M^{0} ×_{𝕋^{c}} ℂP^{ℓ} → S_{Δ}.

The blow-up is encoded by fibrations S_{Δ} → S_{F_{j}} for each F_{j} (see also [5, Proposition 6]): either S_{Δ} = S_{F_{j}}, or the fibration is covered by the obvious projection S̃_{Δ} → ∏_{b≠a_{j}} S_{b} for some index a_{j} such that S_{a_{j}} is a complex projective space and (±g_{a_{j}}, ±ω_{a_{j}}) has constant holomorphic sectional curvature ± ∏_{k≠j}(β_{j} − β_{k})

We unify these cases here by introducing, if S_{Δ} = S_{F_{j}}, an additional index a_{j} with d_{a_{j}} = 0 and S_{a_{j}} = ℂP^{0} (a point). We denote the new index set by Â and take a ∈ Â unless otherwise stated: the additional indices make no difference to the previous formulae. We still have S̃_{Δ} = ∏_{a} S_{a}, and now for all F_{j}, S_{Δ} → S_{F_{j}} is a ℂP^{d_{a_{j}}} -bundle covered by S̃_{Δ} → ∏_{b≠a_{j}} S_{b}. The map j → a_{j} is injective [5] and so Â is the union of a set A and the injective image of {0, … ℓ} (under j ↦ a_{j}). For a ∈ A, either η_{a} < β_{0} or η_{a} > β_{ℓ}, according to the sign of (±g_{a}, ±ω_{a}), whereas for j = {0, … ℓ}, η_{a_{j}} = β_{j}.

The formula (24) for the metric on M^{0} leads to a description [5, Theorem 5] of M̂ as a projective bundle P (L_{0} ⊕ L_{1} ⊕ ⋯ ⊕ L_{ℓ}) → S_{Δ} together with formulae for the Chern classes of L_{j} on the covering S̃_{Δ}. To obtain instead a description of M, we need one further ingredient, which follows easily by considering the form of the covering transformations and the fact that S̃_{Δ} → ∏_{b≠a_{j}} S_{b} covers the fibration S_{Δ} → S_{F_{j}}.

Lemma 10. The projection S̃_{Δ} → S̃ := ∏_{a∈A} S_{a} descends to realize S_{Δ} as a fibre product of flat projective unitary ℂP^{d_{a_{j}}} -bundles over a quotient S of S̃.

An important class of flat projective unitary ℂP^{r}-bundles on S are those of the form P (E), where E is a rank r+1 projectively-flat hermitian vector bundle on S. If S is simply connected, then any flat projective unitary ℂP^{r}-bundle is trivial, hence of the form P (E) with E ≅ E ⊗ ℂ^{r+1} for a holomorphic line bundle E. In general the obstruction to the existence of E is given by a torsion element of H^{2}(S, O^{∗}) (cf. [19]). In particular, such an E always exists if S is a Riemann surface.

Let us suppose that S_{Δ} = P (E_{0}) ×_{S} P (E_{1}) ×_{S} ⋯ ×_{S} P (E_{ℓ}) → S, where each E_{j} → S is projectively-flat hermitian of rank d_{j} + 1. We are free to choose the E_{j} so that M̂ = P (O(−1)_{E_{0}} ⊕ O(−1)_{E_{1}} ⊕ ⋯ ⊕ O(−1)_{E_{ℓ}}) where O(−1)_{E_{j}} is the (fibrewise) tautological line bundle over P (E_{j}) (trivial over the other factors of S_{Δ}). From the description of the blow-up in [5] (see in particular formulae (46) there), we immediately deduce the following (in which we write \overline{c}_{1}(E) = c_{1}(E)/ rank E). M is 𝕋^{c}-equivariantly biholomorphic to P (E_{0} ⊕E_{1} ⋯⊕E_{ℓ}) → S and for any i ≠ j, \overline{c}_{1}(E_{j}) − \overline{c}_{1}(E_{i}) = \frac{1}{2} ∑_{a}(∏_{k≠i}(η_{a} − β_{k}) − ∏_{k≠j}(η_{a} − β_{k}))[ω_{a}/2π].

Derivation of Theorem 3. In order to derive Theorem 3 from the above, it suffices to rescale g and ϕ so that we can take β_{0} = −1 and β_{1} = 1, see [5, Remark 13]. Then we set η_{a} = −1/x_{a} and change the sign of ω_{a} for all a. We also write Ŝ for S_{Δ}, and replace the index set {0, 1} by {0, ∞} so that we can take Â = {0, ∞} ∪ A where A is a finite subset of ℤ^{+}, but these changes are purely cosmetic. A.2. A nonexistence result for order 2 extremal Kähler metrics.

In this paper we study only hamiltonian 2-forms of order 1. As a partial justification for this restriction, we now consider the lowest interesting dimension for the order 2 case, and show that any extremal metric on a compact Kähler 6-manifold compatible with a hamiltonian 2-form of order 2 is a Fubini–Study metric on ℂP^{3}.

According to [4], in this situation the momentum polynomial has nonconstant roots ξ_{1} and ξ_{2} and one constant root η so #A ≤ 1 and p(t) = (t−η)(t−ξ_{1})(t−ξ_{2}). The stable quotient Σ of (M, J) by the complexified 𝕋^{c} action is a compact Riemann surface with Kähler structure (g_{Σ}, ω_{Σ}).

By [5, Remark 13], we can set β_{0} = −1 and β_{2} = 1 and write β_{1} = β (where |β| < 1). If A is empty, (M, J) is biholomorphic to ℂP^{3}; otherwise |η| > 1 and (M, J) is 𝕋^{c}-equivariantly biholomorphic to M = P (L_{0} ⊕ L_{1} ⊕ L_{2}) → Σ, where L_{j} are holomorphic line bundles on Σ such that (without loss) L_{1} is trivial and, by [5, Theorem 5], we have

\begin{aligned}c_{1}(L_{0}) & = \frac{1}{2}(η − 1)(β + 1)[ω_{Σ}/2π], \\ c_{1}(L_{2}) & = \frac{1}{2}(η + 1)(β − 1)[ω_{Σ}/2π]. \end{aligned} ; (25)

The Kähler metric on M is determined by a function Θ(t) satisfying positivity and boundary conditions which imply that Θ(t) = F (t)/(t − η) where F (t) = H(t)((t − η) + H(t)Q(t)) for some function Q(t), and H(t) = (1 − t^{2})(t − β).

If g is extremal and the extremal vector field is tangent to the fibres of M → Σ, then by [4], F (t) is a polynomial of degree at most 5 and g_{Σ} has scalar curvature −F′′(η). This forces Q(t) = 0 and so the scalar curvature of g_{Σ} is 2(3η^{2} − 2βη − 1) which is positive since |η| > 1 and |β| < 1. Hence Σ = ℂP^{1}. Since \frac{1}{4}Scal_{g_{Σ}}[ω_{Σ}/2π] is a primitive integral class, (25) implies that

(η ∓ 1)(β ± 1) = q^{±}(3η^{2} − 2βη − 1)

for some nonzero integers q^{±}. We remark that these formulae show that the relation between q^{±} and (η, β) is birational, in fact the restriction to ℝ^{2} of a quadratic transformation of ℂP^{2}. In any case, η is constant on the lines through (q^{+}, q^{−}) = (1, 1), and β = ±1 on the lines q^{±} = 0 and 2q^{±} − q^{∓} = 1, the latter being the lines on which η = ±1. It follows straightforwardly that |η| > 1 and |β| < 1 iff q^{+} > 0, q^{−} < 0 and 2q^{+} − q^{−} < 1 or vice-versa (swap plus and minus)—which is impossible as |q_{±}| ≥ 1. We therefore have the following nonexistence result.

Theorem 11. A compact extremal Kähler 6-manifold (M, J, g, ω) which admits a hamiltonian 2-form of order 2 with the extremal vector field tangent to the 𝕋^{c}-orbits is isometric to ℂP^{3} with a Fubini–Study metric.

Appendix B. Proof of Lemma 6

In this appendix we prove Lemma 6 by computing the asymptotics as x_{a} → 0, for a ∈ A, of the solution (A, B) of the system (9), i.e., Aα_{1} + Bα_{0} = −2β_{0}, Aα_{2} + Bα_{1} = −2β_{1}, where α_{r} = ∫_{−1}^{1} p_{c}(t)t^{r}dt and β_{r} are as in (7). In order to do this, we rewrite β_{0} and β_{1} as integrals using the obvious identities

p_{c}(1) + p_{c}(−1) = ∫_{−1}^{1} \frac{d}{dt}(tp_{c}(t))dt = ∫_{−1}^{1}(1 + ∑_{a} \frac{d_{a}x_{a}t}{1 + x_{a}t})p_{c}(t)dt

p_{c}(1) − p_{c}(−1) = ∫_{−1}^{1} \frac{d}{dt}(t^{2}p_{c}(t))dt = ∫_{−1}^{1}(2 + ∑_{a} \frac{d_{a}x_{a}t}{1 + x_{a}t})p_{c}(t)t dt

to obtain

β_{0} = ∫_{−1}^{1}(1 + t)^{d_{0}}(1 − t)^{d_{∞}}(1 + d_{0} + d_{∞} + \frac{d_{0}^{2}}{1 + t} + \frac{d_{∞}^{2}}{1 − t} + ∑_{a∈A} \frac{d_{a}x_{a}(s_{a} + t)}{1 + x_{a}t})

× ( ∏_{a∈A}(1 + x_{a}t)^{d_{a}})dt

β_{1} = ∫_{−1}^{1}(1 + t)^{d_{0}}(1 − t)^{d_{∞}}(2 + d_{0} + d_{∞} + \frac{d_{0}^{2}}{1 + t} + \frac{d_{∞}^{2}}{1 − t} + ∑_{a∈A} \frac{d_{a}x_{a}(s_{a} + t)}{1 + x_{a}t})

× ( ∏_{a∈A}(1 + x_{a}t)^{d_{a}})t dt.

The asymptotics of α_{0}, α_{1}, α_{2}, β_{0} and β_{1} are given by integrals of the form

I(m, n, k) = ∫_{−1}^{1}(1 + t)^{m}(1 − t)^{n}t^{k} dt.

Integrating by parts and using 2I(m, n, k + 1) = I(m + 1, n, k) − I(m, n + 1, k),

I(m, n, 0) = \frac{2^{m+n+1} m! n!}{(m + n + 1)!},  I(m, n, 1) = \frac{2^{m+n+1}(m − n) m! n!}{(m + n + 2)!},
I(m, n, 2) = \frac{2^{m+n+1}(m^{2} + n^{2} + m + n − 2mn + 2) m! n!}{(m + n + 3)!}.

These are rather complicated, so we manipulate the integrals using the identities

I(m − 1, n, 0)m^{2} + I(m, n − 1, 0)n^{2} = \frac{1}{2}I(m, n, 0)(m + n + 1)(m + n)
I(m − 1, n, 1)m^{2} + I(m, n − 1, 1)n^{2} = \frac{1}{2}I(m, n, 1)(m + n − 1)(m + n + 2)
I(m − 1, n, 2)m^{2} + I(m, n − 1, 2)n^{2} = \frac{1}{2}I(m, n, 2)(m + n + 3)(m + n)
− I(m, n, 1)(m − n)

and thus obtain, up to O(x^{2}),

α_{k} = I(d_{0}, d_{∞}, k) + I(d_{0}, d_{∞}, k + 1) ∑_{a∈A} d_{a}x_{a},

β_{0} = \frac{1}{2}I(d_{0}, d_{∞}, 0)(1 + d_{0} + d_{∞})(2 + d_{0} + d_{∞}) + I(d_{0}, d_{∞}, 0) ∑_{a∈A} d_{a}s_{a}x_{a}

+ \frac{1}{2}I(d_{0}, d_{∞}, 1)(1 + d_{0} + d_{∞})(2 + d_{0} + d_{∞}) ∑_{a∈A} d_{a}x_{a},

β_{1} = \frac{1}{2}I(d_{0}, d_{∞}, 1)(1 + d_{0} + d_{∞})(2 + d_{0} + d_{∞}) + I(d_{0}, d_{∞}, 1) ∑_{a∈A} d_{a}s_{a}x_{a}

+ (\frac{1}{2}I(d_{0}, d_{∞}, 2)(3 + d_{0} + d_{∞})(2 + d_{0} + d_{∞}) − I(d_{0}, d_{∞}, 1)(d_{0} − d_{∞})) ∑_{a∈A} d_{a}x_{a}.

Direct computation with these formulae and the identity I(m, n, 1)(m + n + 2) = I(m, n, 0)(m − n) now shows that, up to O(x^{2}),

\frac{α_{0}β_{1} − α_{1}β_{0}}{α_{0}α_{2} − α_{1}^{2}} = (2 + d_{0} + d_{∞}) ∑_{a∈A} d_{a}x_{a}

\frac{α_{2}β_{0} − α_{1}β_{1}}{α_{0}α_{2} − α_{1}^{2}} = \frac{1}{2}(1 + d_{0} + d_{∞})(2 + d_{0} + d_{∞}) + ∑_{a∈A} d_{a}s_{a}x_{a} + (d_{∞} − d_{0}) ∑_{a∈A} d_{a}x_{a}.

Multiplying by −2 completes the proof.

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Vestislav Apostolov, Département de Mathématiques, UQAM, C.P. 8888, Succ. Centre-ville, Montréal (Québec), H3C 3P8, Canada, and, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. bl. 8, 1113 Sofia, Bulgaria

   E-mail address: apostolov.vestislav@uqam.ca

David M. J. Calderbank, Department of Mathematics, University of York, Heslington, York YO10 5DD, England

   E-mail address: dc511@york.ac.uk

Paul Gauduchon, Centre de Mathématiques, École Polytechnique, UMR 7640 du CNRS, 91128 Palaiseau, France

   E-mail address: pg@math.polytechnique.fr

Christina W. Tønnesen-Friedman, Department of Mathematics, Union College, Schenectady, New York 12308, USA

   E-mail address: tonnesec@union.edu